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College Algebra (Kaufmann), 8th ed.

1. 6 64x3 2 48x2 1 12x 2 1 7 9x3y 1 12x4y2 8 3x 2x 1 1 3x 2 4 9 5x 1 2 6x 2 5 10 8 x 1 2 x2 2 2x 1 4 11 x 2 2 x 1 y 12 21x5 20 13 x 1 3 x2 1 2x 1 4 14 n 2 8 12 15 23x 1 6 6x x 2 3 x 1 2 16 8 2 13n 2n2 17 2y2 2 5xy 3y2 1 4x 18 12x227x 19 522 6 20 423 1 322 5 21 2xy2 3 6xy2 22 24 2 3i 23 34 2 18i 24 85 1 0i 25 1 10 1 7 10 i 90360_ANS1_A1 A44 indd 4 11 17 11 11 19 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This page contains answers for this chapter only
2. 37 7x 24 39 35b 1 12a3 80a2b2 41 12 1 9n 2 10n2 12n2 43 9y 1 8x 2 12xy 12xy 45 13x 1 14 2x 1 1 3x 1 4 47 7x 1 21 x x 1 7 49 1 a 2 2 51 5 2 x 2 1 53 2n 1 10 3 n 1 1 n 2 1 55 1 x 1 1 57 9x 1 73 x 1 3 x 1 7 x 1 9 59 3x2 1 30x 2 78 x 1 1 x 2 1 x 1 8 x 2 2 61 x 1 6 x 2 3 2 63 2x2 2 x 1 1 x 1 1 x 2 1 65 28 n2 1 4 n 1 2 n 2 2 67 5x2 1 16x 1 5 x 1 1 x 2 4 x 1 7 69 a 5 x 2 1 c 5 a 2 3 e x 1 3 71 5y2 2 3xy2 x2y 1 2x2 73 x 1 1 x 2 1 75 n 2 1 n 1 1 77 26x 2 4 3x 1 9 83 y 2 x2 x2y 85 3b3 1 2a2 a2b3 87 y2 2 x2 xy 89 12x3a11 91 1 93 x2a 95 24y6b12 97 xb 99 6 2 10 7 101 4 12 10 24 103 180 000 105 0 0000023 107 0 04 109 30 000 111 0 03 117 a 4 385 1014
3. The Use of sets Some of the vocabulary and symbolism associated with the concept of sets can be ef fectively used in the study of algebra A set is a collection of objects the objects are called elements or members of the set The use of capital letters to name sets and the use of set braces 5 6 to enclose the elements or a description of the elements provide a convenient way to communicate about sets For example a set A that consists of the vowels of the English alphabet can be represented as follows A 5 vowels of the English alphabet Word description or A 5 a e i o u List or roster description or A 5 x 0 x is a vowel Set builder notation A set consisting of no elements is called the null set or empty set and is written Set builder notation combines the use of braces and the concept of a variable For example x 0 x is a vowel is read the set of all x such that x is a vowel Note that the vertical line is read such that Two sets are said to be equal if they contain exactly the same elements For ex ample 1 2 3 5 2 1 3 because both sets contain exactly the same elements the order in which the elements are listed does not matter A slash mark through an equal ity symbol denotes not equal to Thus if A 5 1 2 3 and B 5 3 6 we can write A B which is read set A is not equal to set B Real Numbers The following terminology is commonly used to classify different types of num
4. a 0 00063 960 000 3200 0 0000021 b 290 000 solution a 0 00063 960 000 3200 0 0000021 5 6 3 1024 9 6 105 3 2 103 2 1 1026 5 6 3 9 6 101 3 2 2 1 1023 5 9 104 5 90 000 b 290 000 5 2 9 104 5 292104 5 3 102 5 3 100 5 300 Many calculators are equipped to display numbers in scienti c notation The dis play panel shows the number between 1 and 10 and the appropriate exponent of 10 For example evaluating 3 800 000 2 yields 1 444E13 Thus 3 800 000 2 5 1 444 1013 5 14 440 000 000 000 Similarly the answer for 0 000168 2 is displayed as 2 8224E28 Thus 0 000168 2 5 2 8224 1028 5 0 000000028224 Calculators vary in the number of digits they display between 1 and 10 when they represent a number in scienti c notation For example we used two different calcula tors to estimate 6729 6 and obtained the following results 9 283316768E22 9 28331676776E22 Obviously you need to know the capabilities of your calculator when working with problems in scienti c notation Many calculators also allow you to enter a number in scienti c notation Such calculators are equipped with an enter the exponent key often labeled EE Thus a number such as 3 14 108 might be entered as follows enter Press Display 3 14 EE 3 14E 8 3 14E8 classroom example Use scientific notation to perform the indicated operations a 0 0048 20 000 0 000
5. 55 1026 1 8230 57 3x26y 2 622xy 59 13 1 723 61 30 1 1126 63 16 65 x 1 22xy 1 y 67 a 2 b 69 325 2 6 71 27 1 23 73 2 2210 1 3214 43 75 x 1 2x x 2 1 77 x 2 2xy x 2 y 79 6x 1 72xy 1 2y 9x 2 4y 81 2 22x 1 2h 1 22x 83 1 2x 1 h 2 3 1 2x 2 3 91 4x2 93 y223y 95 2m427 97 3d322d 99 4n1025 Problem Set 0 7 page 79 1 7 3 8 5 24 7 2 9 64 11 0 001 13 1 32 15 2 17 15x712 19 y512 21 64x34y32 23 4x415 25 7 a112 27 16x43 81y 29 y32 x 31 8a92x2 33 2 4 8 35 2 12 x7 37 xy2 4 xy3 39 a2 12 a5b11 41 42 6 2 43 2 6 2 45 22 47 x2 12 x7 49 52 3 x2 x 51 2 6 x3y4 y 53
6. 58 3x 2 2 4x 1 3 59 2 6x 1 3h x2 x 1 h 2 60 12 x2 1 2 32 61 2023 62 6x26x 63 2xy2 3 4xy2 64 23 65 210x 2y 66 15 2 322 23 67 24 2 426 15 68 3x 1 62xy x 2 4y 69 2 6 55 70 2 12 x11 71 x22 6 x5 72 x2 10 xy9 73 2 6 5 74 2 12 x11 x 75 211 2 6i 76 21 2 2i 77 1 2 2i 78 21 1 0i 79 26 2 7i 80 225 1 15i 81 214 2 12i 82 29 1 0i 83 0 2 5 3 i 84 2 6 25 1 17 25 i 85 0 1 i 86 2 12 29 2 30 29 i 87 10i 88 2i210 89 16i25 90 212 91 2423 92 222 93 600 000 000 94 800 000 chapter 0 Test page 102 1 a 2 1 49 b 8 27 c 8 27 d 3 4 2 2 15 x4y2 3 212x 2 8 4 230x 2 1 32x 2 8 5 3x3 1 4x2 2 11x 2 14
7. For Problems 51 70 find each product and express the answers in standard form Objective 3 51 3i 7i 52 25i 8i 53 4i 3 2 2i 54 5i 2 1 6i 55 3 1 2i 4 1 6i 56 7 1 3i 8 1 4i 57 4 1 5i 2 2 9i 58 1 1 i 2 2 i 59 22 2 3i 4 1 6i 60 23 2 7i 2 1 10i 61 6 2 4i 21 2 2i 62 7 2 3i 22 2 8i 63 3 1 4i 2 64 4 2 2i 2 65 21 2 2i 2 66 22 1 5i 2 67 8 2 7i 8 1 7i 68 5 1 3i 5 2 3i 69 22 1 3i 22 2 3i 70 26 2 7i 26 1 7i For Problems 71 84 find each quotient and express the answers in standard form Objective 3 71 4i 3 2 2i 72 3i 6 1 2i 73 2 1 3i 3i 74 3 2 5i 4i 75 3 2i 76 7 4i 77 3 1 2i 4 1 5i 78 2 1 5i 3 1 7i 79 4 1 7i 2 2 3i 80 3 1 9i 4 2 i 81 3 2 7i 22 1 4i 82 4 2 10i 23 1 7i 83 21 2 i 22 2 3i 84 24 1 9i 23 2 6i 85 Using a 1 bi and c 1 di to represent two complex numbers verify the following properties a The conjugate of the sum of two complex num bers is equal to the sum of the conjugates of the two numbers b The conjugate of the product of two complex numbers is equal to the product of the conju gates of the numbers 86 Is every real number also a complex number Explain your answer 87 Can the product of two nonreal comp
8. 1 1 or x 1 1 3x 1 2 By checking the middle term formed in each of these products we find that the second possibility yields the desired middle term of 5x Therefore 3x 2 1 5x 1 2 5 x 1 1 3x 1 2 eXAMPLe 5 Factor 8x 2 2 30xy 1 7y2 solution First observe that the first term 8x 2 can be written as 2x 4x or x 8x Second be cause the middle term is negative and the last term is positive we know that the binomi als are of the form 2x 2 __ 4x 2 __ or x 2 __ 8x 2 __ Third because the factors of the last term 7y 2 are 1y and 7y the following possibilities exist 2x 2 1y 4x 2 7y 2x 2 7y 4x 2 1y x 2 1y 8x 2 7y x 2 7y 8x 21y By checking the middle term formed in each of these products we find that 2x 2 7y 4x 2 1y produces the desired middle term of 230xy Therefore 8x 2 2 30xy 1 7y 2 5 2x 2 7y 4x 2 y eXAMPLe 6 Factor 10x 2 2 36x 2 16 solution First note that there is a common factor of 2 By using the distributive property we obtain 10x2 2 36x 2 16 5 2 5x2 2 18x 2 8 Now let s determine if 5x2 2 18x 2 8 can be factored The first term 5x2 can be written as x 5x The last term 28 can be written as 22 4 2 24 21 8 or 1 28 Therefore we have the following possibilities to try x 2 2 5x 1 4 x 1
9. 16 39 41 50 1 26 43 a 18 c 39 e 35 45 Commutative property of multiplication 47 Identity property of multiplication 49 Multiplication property of negative one 51 Distributive property 53 Commutative property of multiplication 55 Distributive property 57 Associative property of multiplication 59 222 61 100 63 221 65 8 67 19 69 66 71 275 73 34 75 1 77 11 79 4 81 x y 4 1 83 x y 0 3 85 x y 5 1 87 Quadrant IV 89 Quadrant III 91 Quadrant I Problem Set 0 2 page 28 1 1 8 3 2 1 1000 5 27 7 4 9 227 8 11 1 13 16 25 15 4 17 1 100 or 0 01 19 1 100 000 or 0 00001 21 81 23 1 16 25 3 4 27 256 25 29 16 25 31 64 81 33 64 35 1 100 000 or 0
10. 2 x 2 23 x 2 1 7x 2 36 24 x 2 2 4xy 2 5y 2 25 3x 2 2 11x 1 10 26 2x 2 2 7x 2 30 27 10x 2 1 17x 1 7 28 8y 2 1 22y 2 21 29 10x 2 1 39x 2 27 30 3x 2 1 x 2 5 31 36a2 2 12a 1 1 32 18n3 1 39n2 2 15n 33 8x 2 1 2xy 2 y2 34 12x 2 1 7xy 2 10y 2 35 2n2 2 n 2 5 36 6x 2 2 x 2 12 For Problems 37 40 factor the sum or difference of two cubes Objective 5 37 x 3 2 8 38 x 3 1 64 39 64x 3 1 27y 3 40 27x 3 2 8y 3 For Problems 41 66 factor each polynomial com pletely Indicate any that are not factorable using inte gers Objective 6 41 4x 4 1 16 42 n3 2 49n 43 x 3 2 9x 44 12n2 1 59n 1 72 45 9a 2 2 42a 1 49 46 1 2 16x 4 47 2n 3 1 6n 2 1 10n 48 25t 2 2 100 49 2n 3 1 14n 2 2 20n 50 25n 2 1 64 51 4x 3 1 32 52 2x 3 2 54 53 x 4 2 4x 2 2 45 54 x 4 2 x 2 2 12 55 2x4y 2 26x2y 2 96y 56 3x4y 2 15x2y 2 108y 57 a 1 b 2 2 c 1 d 2 58 a 2 b 2 2 c 2 d 2 59 x 2 1 8x 1 16 2 y 2 60 4x 2 1 12x 1 9 2 y 2 61 x 2 2 y 2 2 10y 2 25z 62 y 2 2 x 2 1 16x 2 64 63 60x 2 2 32x 2 15 64 40x 2 1 37x 2 63 65 84x 3 1 57x 2 2 60x 66 210x3 2 102x2 2180x 90360_ch0_001 102
11. 86 plot the following points on a rectangular coordinate system Objective 7 81 24 1 82 3 22 83 0 23 84 22 22 85 5 21 86 1 4 For Problems 87 92 state the quadrant that contains the point Objective 7 87 4 22 88 23 1 89 26 22 90 5 2 91 1 8 92 27 27 54 x 1 2 1 22 5 x 1 2 1 22 55 6 4 1 7 4 5 6 1 7 4 56 a2 3ba3 2b 5 1 57 4 5x 5 4 5 x 58 17 8 25 5 17 8 25 For Problems 59 80 evaluate each of the algebraic expressions for the given values of the variables Objective 6 59 5x 1 3y x 5 22 and y 5 24 60 7x 2 4y x 5 21 and y 5 6 61 23ab 2 2c a 5 24 b 5 7 and c 5 28 62 x 2 2y 1 3z x 5 23 y 5 24 and z 5 9 63 a 2 2b 1 3c 2 4 a 5 6 b 5 25 and c 5 211 64 3a 2 2b 2 4c 1 1 a 5 4 b 5 6 and c 5 28 65 22x 1 7y x 2 y x 5 23 and y 5 22 66 x 2 3y 1 2z 2x 2 y x 5 4 y 5 9 z 5 212 67 5x 2 2y 23x 1 4y x 5 23 and y 5 2 7 68 2a 2 7b 4a 1 3b a 5 6 and b 5 23 69 5x 1 4y 2 9y 2 2y x 5 2 and y 5 28 70 5a 1 7b 2 9a 2 6b a 5 27 and b 5 8 93 Do you think 322 is a rational or an irrational number Defend your answer 94 Explain why 0 8 5 0 but 8 0 is unde ned 95 The solution of the following simpli cation prob lem is incorrect The answer should be 211 Find and correct the error 8 4 24 2 2 3 4 4 2 1 21 5 22 2 2 12 4 1 85
12. Note that b is a real number even though it is called the imaginary part Each of the following represents a complex number 6 1 2i is already expressed in the form a 1 bi Traditionally complex numbers for which a 0 and b 0 have been called imaginary numbers 5 2 3i can be written 5 1 23i even though the form 5 2 3i is often used 28 1 i12 can be written 28 1 12i It is easy to mistake 12i for 12i Thus we commonly write i12 instead of 12i to avoid any difficulties with the radical sign 29i can be written 0 1 29i Complex numbers such as 29i for which a 0 and b 0 traditionally have been called pure imaginary numbers 5 can be written 5 1 0i 0 8 Complex Numbers O B j e c T i V e s 1 Express the square root of a negative number in terms of i 2 Add and subtract complex numbers 3 Multiply and divide complex numbers 90360_ch0_001 102 indd 82 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 8 Complex Number
13. b 1 2 i 3 c 1 2 2i 3 d 1 1 i 4 e 2 2 i 4 f 21 1 i 5 Answers to the concept Quiz 1 True 2 True 3 False 4 True 5 True 6 True 7 False 8 True 90360_ch0_001 102 indd 90 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User OBjecTiVe Recognize the vocabulary and symbolism associated with sets section 0 1 Objective 1 sUMMARy Be sure of the following key concepts about sets Elements null set equal sets subsets and set builder notation eXAMPLe Answer True or False a a b c 5 b a c b 51 3 56 50 1 2 3 4 56 solution a True b True 91 Chapter 0 sUMMARy continued Know the various subset classifications of the real number system section 0 1 Objective 2 The sets of natural numbers whole numbers integers ra tional numbers and irrational numbers are all subsets of the real number system Name each of the following sets a 0 1 2 3 b 5 p 23 22 21 06 c 51 2 3 p 6 solution a Whole nu
14. d 8619 6 e 314 5 f 145 723 2 Thoughts into Words Graphing calculator Activities 90360_ch0_001 102 indd 30 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 3 Polynomials 31 118 Use your calculator to estimate each of the follow ing Express nal answers in ordinary notation rounded to the nearest one thousandth a 1 09 5 b 1 08 10 c 1 14 7 d 1 12 20 e 0 785 4 f 0 492 5 Answers to the concept Quiz 1 True 2 False 3 False 4 False 5 False 6 False 7 False 8 False 9 True 10 False 0 3 Polynomials O B j e c T i V e s 1 Add and subtract polynomials 2 Multiply polynomials 3 Perform binomial expansions 4 Divide a polynomial by a monomial Recall that algebraic expressions such as 5x 26y 2 2x 21y 22 14a 2b 5x 24 and 217ab 2c 3 are called terms Terms that contain variables with only nonnegative inte gers as exponents are called monomials Of the previously listed terms 5x 26y 2 14a 2b and 217ab 2c 3 are monomial
15. n 2 1 5 an 2 1 n 2 1ba1 1b 2 n2 n 2 1 5 n 2 1 n 2 1 2 n2 n 2 1 5 n 2 1 2 n2 n 2 1 or 2n2 1 n 2 1 n 2 1 Finally we need to recognize that complex fractions are sometimes the result of applying the definition b2n 5 1 bn Our final example illustrates this idea eXAMPLe 10 Simplify 2x 21 1 y 21 x 2 3y22 solution First let s apply b 2n 5 1 bn 2x21 1 y21 x 2 3y22 5 2 x 1 1 y x 2 3 y2 Now we can proceed as in the previous examples 2 x 1 1 y x 2 3 y2 axy2 xy2b 5 2 x xy2 1 1 y xy2 x xy2 2 3 y2 xy2 5 2y2 1 xy x2y2 2 3x classroom example Simplify 4 2 y 1 2 4 y classroom example Simplify 5a22 1 b21 a 1 2b21 90360_ch0_001 102 indd 60 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 61 For Problems 1 7 answer true or false 1 The indicated quotient of two polynomials is called a ration
16. t forget to simplify the final result n2 n 2 1 2 1 n 2 1 5 n2 2 1 n 2 1 5 n 1 1 n 2 1 n 2 1 5 n 1 1 If we need to add or subtract rational expressions that do not have a common de nominator then we apply the property a gt b 5 a k gt b k to obtain equivalent frac tions with a common denominator Study the next examples and again pay special at tention to the format we used to organize our work 90360_ch0_001 102 indd 54 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 55 Remark Remember that the least common multiple of a set of whole numbers is the smallest nonzero whole number divisible by each of the numbers in the set When we add or subtract rational numbers the least common multiple of the denominators of those numbers is the least common denominator LCD This concept of a least common denominator can be extended to include polynomials eXAMPLe 1 Add x 1 2 4 1 3x 1 1 3 solution By inspection we
17. x 3 2 y 3 Thus we can state the following factoring patterns x 3 1 y 3 5 x 1 y x 2 2 xy 1 y 2 x 3 2 y 3 5 x 2 y x 2 1 xy 1 y 2 Note how these patterns are used in the next three examples x 3 1 8 5 x 3 1 2 3 5 x 1 2 x 2 2 2x 1 4 8x 3 2 27y 3 5 2x 3 2 3y 3 5 2x 2 3y 4x 2 1 6xy 1 9y 2 8a 6 1 125b 3 5 2a 2 3 1 5b 3 5 2a 2 1 5b 4a 4 2 10a 2b 1 25b 2 Applying More Than One Factoring Technique We do want to leave you with one final word of caution Be sure to factor completely Sometimes more than one technique needs to be applied or perhaps the same technique can be applied more than once Study the following examples very carefully 90360_ch0_001 102 indd 48 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 4 Factoring Polynomials 49 2x 2 2 8 5 2 x 2 2 4 5 2 x 1 2 x 2 2 3x 2 1 18x 1 24 5 3 x 2
18. x2 1 2 1y2 2 For Problems 61 68 express each in simplest radical form All variables represent positive real numbers 61 5248 62 3224x3 63 2 3 32x4y5 64 328 226 65 B 5x 2y2 66 3 22 1 5 67 422 322 1 23 68 32x 2x 2 22y For Problems 69 74 perform the indicated operations and express the answers in simplest radical form 69 252 3 5 70 2 3 x22 4 x 71 2x32 3 x4 72 2xy2 5 x3y2 73 25 2 3 5 74 2 3 x2 2 4 x3 For Problems 75 86 perform the indicated operations and express the resulting complex number in standard form 75 27 1 3i 1 24 2 9i 76 2 2 10i 2 3 2 8i 77 21 1 4i 2 22 1 6i 78 3i 27i 79 2 2 5i 3 1 4i 80 23 2 i 6 2 7i 81 4 1 2i 24 2 i 82 5 2 2i 5 1 2i 83 5 3i 84 2 1 3i 3 2 4i 85 21 2 2i 22 1 i 86 26i 5 1 2i For Problems 87 92 write each in terms of i and simplify 87 22100 88 2240 89 42280 90 A 229BA 2216B 91 A 226BA 228B 92 2224 223 For Problems 93 and 94 use scienti c notation and the properties of exponents to help with the computations 93 0 0064 420 000 0 00014 0 032 94 8600 0 0000064 0 0016 0 000043 Chapter 0 Review Problem Set 90360_ch0_001 102 indd 101 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some t
19. 1 3i 1 4 1 5i 3 8 1 6i 2 5 1 2i 4 26 1 4i 2 4 1 6i 5 27 2 3i 1 24 1 4i 6 6 2 7i 2 7 2 6i 7 22 2 3i 2 21 2 i 8 a1 3 1 2 5 ib 1 a1 2 1 1 4 ib 9 a23 4 2 1 4 ib 1 a3 5 1 2 3 ib 10 a5 8 1 1 2 ib 2 a7 8 1 1 5 ib 11 a 3 10 2 3 4 ib 2 a2 2 5 1 1 6 ib 12 A4 1 i23B 1 A26 2 2i23B 13 5 1 3i 1 7 2 2i 1 28 2 i 14 5 2 7i 2 6 2 2i 2 21 2 2i For Problems 15 30 write each in terms of i and sim plify Objective 1 For example 2220 5 i220 5 i2425 5 2i25 15 229 16 2249 17 2219 18 2231 19 B2 4 9 20 B2 25 36 21 228 22 2218 23 2227 24 2232 25 2254 26 2240 27 32236 28 52264 29 42218 30 6228 Problem set 0 8 90360_ch0_001 102 indd 88 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 8 Complex Numbers 89
20. 1 a 5 0 For every nonzero real number a there exists a unique real number 1 a such that a a1 ab 5 1 a a 5 1 Multiplication property a 0 5 0 a 5 0 of zero Multiplication property a 21 5 21 a 5 2a of negative one Distributive property a b 1 c 5 ab 1 ac Let s make a few comments about the properties of real numbers The set of real numbers is said to be closed with respect to addition and multiplication That is the sum of two real numbers is a real number and the product of two real numbers is a real number Closure plays an important role when we are proving additional properties that pertain to real numbers Addition and multiplication are said to be commutative operations This means that the order in which you add or multiply two real numbers does not affect the result For example 6 1 28 5 28 1 6 and 24 23 5 23 24 It is important to real ize that subtraction and division are not commutative operations order does make a difference For example 3 2 4 5 21 but 4 2 3 5 1 Likewise 2 4 1 5 2 but 1 4 2 5 1 2 Addition and multiplication are associative operations The associative properties are grouping properties For example 28 1 9 1 6 5 28 1 9 1 6 changing the grouping of the numbers does not affect the final sum Likewise for multiplication 24 23 2 5 24 23 2 Subtraction and division
21. 2 4 48x5 28 2 4 162x6y7 29 B 12 25 30 B 75 81 For Problems 31 44 rationalize the denominator and express the result in simplest radical form Objective 5 31 B 7 8 32 235 27 33 426 210 34 227 218 35 623 726 36 B 3x 2y 37 25 212x4 38 25y 218x3 39 212a2b 25a3b3 40 5 2 3 3 41 2 3 27 2 3 4 42 B 3 5 2x 43 2 3 2y 2 3 3x 44 2 3 12xy 2 3 3x2y5 For Problems 45 52 use the distributive property to help simplify each Objective 3 For example 328 1 522 5 32422 1 522 5 622 1 522 5 6 1 5 22 5 1122 45 5212 1 223 46 4250 2 9232 47 2228 2 3263 1 827 48 42 3 2 1 22 3 16 2 2 3 54 49 5 6248 2 3 4212 50 2 5240 1 1 6290 51 228 3 2 3218 5 2 250 2 52 32 3 54 2 1 52 3 16 3 6 2 23 2 5 23 7 If x 0 then 2x2 5 2x 8 For real numbers the process of rationalizing the denominator changes the de nominator from an irrational number to a rational number Problem set 0 6 90360_ch0_001 102 indd 73 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the
22. 28 solution 26 28 5 B 6 8 Remember that 2a 2b 5 B a b 5 B 3 4 Reduce the fraction 5 23 24 5 23 2 eXAMPLe 3 Simplify 2 3 5 2 3 9 solution 2 3 5 2 3 9 5 2 3 5 2 3 9 2 3 3 2 3 3 5 2 3 15 2 3 27 5 2 3 15 3 classroom example Simplify 27 218 classroom example Simplify 26 227 classroom example Simplify 2 3 7 2 3 2 90360_ch0_001 102 indd 70 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 6 Radicals 71 Now let s consider an example in which the denominator is of binomial form eXAMPLe 4 Simplify 4 25 1 22 by rationalizing the denominator solution Remember that a moment ago we found that A 25 1 22BA 25 2 22B 5 3 Let s use that idea here 4 25 1 22 5 a 4 25 1 22 ba 25 2 22 25 2 22 b 5 4A25 2 22B A25 1 22BA25 2 22B 5 4A25 2 22B 3 The process of rationalizing the denominator does agree with the
23. 30a5 2 24a3 1 54a2 26a 70 18x 3y 2 1 27x 2y 3 3xy 71 220a3b2 2 44a4b5 24a2b 72 21x 5y 6 1 28x 4y 3 2 35x 5y 4 7x 2y 3 For Problems 73 82 nd the indicated products Assume all variables that appear as exponents represent integers Objectives 2 and 3 73 x a 1 y b x a 2 y b 74 x 2a 1 1 x 2a 2 3 75 x b 1 4 x b 2 7 76 3x a 2 2 x a 1 5 77 2x b 2 1 3x b 1 2 78 2x a 2 3 2x a 1 3 79 x 2a 2 1 2 80 x 3b 1 2 2 81 x a 2 2 3 82 x b 1 3 3 90360_ch0_001 102 indd 39 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 40 83 Describe how to multiply two binomials 84 Describe how to multiply a binomial and a trino mial 85 Determine the number of terms in the product of x 1 y and a 1 b 1 c 1 d without doing the multiplication Explain how you arrived at your answer 86 Use the computing feature of your graphing calcu lator to check at least one real number for yo
24. 4 5x 2 2 x 2 1 5x 1 8 x 1 8 5x 2 1 x 1 2 5x 2 4 x 2 4 5x 1 2 x 1 1 5x 2 8 x 2 8 5x 1 1 By checking the middle terms we find that x 2 4 5x 1 2 yields the desired middle term of 218x Thus 10x 2 2 36x 2 16 5 2 5x 2 2 18x 2 8 5 2 x 2 4 5x 1 2 classroom example Factor 5a2 1 8a 1 3 classroom example Factor 6x2 1 17xy 1 5y2 classroom example Factor 6a2 1 46a 1 28 90360_ch0_001 102 indd 45 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 46 eXAMPLe 7 Factor 4x 2 1 6x 1 9 solution The first term 4x 2 and the positive signs of the middle and last terms indicate that the binomials are of the form x 1 __ 4x 1 __ or 2x 1 __ 2x 1 __ Because the factors of the last term 9 are 1 and 9 or 3 and 3 we have the follow ing possibilities to try x 1 1 4x 1
25. 5 4y 2 1 3y 2 1 2y 1 7 1 2 5 7y 2 2 y 1 9 Multiplying Polynomials The distributive property is usually stated as a b1c 5 ab1ac but it can be extended as follows a b 1 c 1 d 5 ab 1 ac 1 ad a b 1 c 1 d 1 e 5 ab 1 ac 1 ad 1 ae etc The commutative and associative properties the properties of exponents and the distributive property work together to form the basis for finding the product of a mono mial and a polynomial with more than one term The following example illustrates this idea 3x2 2x2 1 5x 1 3 5 3x2 2x2 1 3x2 5x 1 3x2 3 5 6x4 1 15x3 1 9x2 90360_ch0_001 102 indd 32 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 3 Polynomials 33 Extending the method of nding the product of a monomial and a polynomial to nding the product of two polynomials each of which has more than one term is again based on the distributive property x 1 2 y 1 5 5 x y 1 5 1 2 y 1 5 5 x y 1 x 5 1 2 y 1 2 5 5 xy 1 5x 1 2y 1 10 In the nex
26. Some Basic Concepts of Algebra A Review 20 We especially want to call your attention to the last example in each column Note that 25 2 means that 25 is the base used as a factor twice However 25 2 means that 5 is the base and after it is squared we take the opposite of the result Properties of exponents In a previous algebra course you may have seen some properties pertaining to the use of positive integers as exponents Those properties can be summarized as follows Property 0 1 Properties of exponents If a and b are real numbers and m and n are positive integers then 1 bn bm 5 bn 1 m 2 bn m 5 bmn 3 ab n 5 anbn 4 aa bb n 5 an bn b 0 5 bn bm 5 bn 2 m when n m b 0 bn bm 5 1 when n 5 m b 0 bn bm 5 1 bm2n when n m b 0 Each part of Property 0 1 can be justi ed by using De nition 0 2 For example to jus tify part 1 we can reason as follows bn bm 5 bbb p b bbb p b 14243 14243 n factors m factors of b of b 5 bbb p b 14243 n 1 m factors of b 5 bn1m Similar reasoning can be used to verify the other parts of Property 0 1 The following examples illustrate the use of Property 0 1 along with the commutative and associative properties of the real numbers We have chosen to show all of the steps however many of the steps can be performed mentally 90360_ch0_001 102 indd 20 11 17 11 8 47 AM Copyright 2012 Ceng
27. This ap proach works effectively with complex fractions when the LCD of all the denomina tors is easy to find Let s look at a type of complex fraction used in certain calculus problems eXAMPLe 8 Simplify 1 x 1 h 2 1 x h solution 1 x 1 h 2 1 x h 1 5 c x x 1 h x x 1 h d 1 x 1 h 2 1 x h 1 5 x x 1 h a 1 x 1 hb 2 x x 1 h a1 xb x x 1 h h 5 x 2 x 1 h hx x 1 h 5 x 2 x 2 h hx x 1 h 5 2h hx x 1 h 5 2 1 x x 1 h classroom example Simplify 3 x 1 h 2 3 x h 90360_ch0_001 102 indd 59 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 60 Example 9 illustrates another way to simplify complex fractions eXAMPLe 9 Simplify 1 2 n 1 2 1 n solution We first simplify the complex fraction by multiplying by n gt n n 1 2 1 n an nb 5 n2 n 2 1 Now we can perform the subtraction 1 2 n2
28. c 2 322 1017 e 3 052 1012 Problem Set 0 3 page 38 1 14x2 1 x 2 6 3 2x2 2 4x 2 9 5 6x 2 11 7 6x2 2 5x 2 7 9 2x 2 34 11 12x3y2 1 15x2y3 13 30a4b3 2 24a5b3 1 18a4b4 15 x2 1 20x 1 96 17 n2 2 16n 1 48 19 sx 1 sy 2 tx 2 ty 21 6x2 1 7x 2 3 23 12x2 2 37x 1 21 25 x2 1 8x 1 16 27 4n2 1 12n 1 9 29 x3 1 x2 2 14x 2 24 31 6x3 2 x2 2 11x 1 6 33 x3 1 2x2 2 7x 1 4 35 t3 2 1 37 6x3 1 x2 2 5x 2 2 39 x4 1 8x3 1 15x2 1 2x 2 4 41 25x2 2 4 43 x4 2 10x3 1 21x2 1 20x 1 4 45 4x2 2 9y2 47 x3 1 15x2 1 75x 1 125 49 8x3 1 12x2 1 6x 1 1 51 64x3 2 144x2 1 108x 2 27 53 125x3 2 150x2y 1 60xy2 2 8y3 55 a7 1 7a6b 1 21a5b2 1 35a4b3 1 35a3b4 1 21a2b5 1 7ab6 1 b7 57 x5 2 5x4y 1 10x3y2 2 10x2y3 1 5xy4 2 y5 59 x4 1 8x3y 1 24x2y2 1 32xy3 1 16y4 61 64a6 2 192a5b 1 240a4b2 2 160a3b3 1 60a2b4 2 12ab5 1 b6 63 x14 1 7x12y 1 21x10y2 1 35x8y3 1 35x6y4 1 21x4y5 1 7x
29. i a 1 bi 2 c 1 di 5 a 2 c 1 b 2 d i Add the complex numbers 3 2 6i 1 27 2 3i solution 3 2 6i 1 27 2 3i 5 3 2 7 1 26 2 3 i 5 24 2 9i Multiply and divide com plex numbers section 0 8 Objective 3 The product of two complex numbers follows the same pat tern as the product of two bino mials When simplifying replace any i2 with 21 To simplify expressions that in dicate the quotient of complex numbers like 4 1 3i 5 2 2i multiply the numerator and denominator by the conjugate of the denomi nator The conjugate of a 1 bi is a 2 bi The product of a complex number and its conju gate is a real number Find the quotient 2 1 3i 4 2 i and express the answer in standard form of a complex number solution Multiply the numerator and denominator by 4 1 i the conjugate of the denominator 2 1 3i 4 2 i 5 2 1 3i 4 2 i 4 1 i 4 1 i 5 8 1 14i 1 3i2 16 2 i2 5 8 1 14i 1 3 21 16 2 21 5 5 1 14i 17 5 5 17 1 14 17i Chapter 0 Summary 90360_ch0_001 102 indd 99 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right
30. x 2 x2 x 1 7 2 x x2 2 49 65 2n2 n4 2 16 2 n n2 2 4 1 1 n 1 2 66 n n2 1 1 1 n2 1 3n n4 2 1 2 1 n 2 1 67 2x 1 1 x2 2 3x 2 4 1 3x 2 2 x2 1 3x 2 28 68 3x 2 4 2x2 2 9x 2 5 2 2x 2 1 3x2 2 11x 2 20 25 5xy x 1 6 x2 2 36 x2 2 6x 26 2a2 1 6 a2 2 a a3 2 a2 8a 2 4 27 5a2 1 20a a3 2 2a2 a2 2 a 2 12 a2 2 16 28 t4 2 81 t2 2 6t 1 9 6t2 2 11t 2 21 5t2 1 8t 2 21 29 x2 1 5xy 2 6y2 xy2 2 y3 2x2 1 15xy 1 18y2 xy 1 4y2 30 10n2 1 21n 2 10 5n2 1 33n 2 14 2n2 1 6n 2 56 2n2 2 3n 2 20 31 9y2 x2 1 12x 1 36 4 12y x2 1 6x 32 x2 2 4xy 1 4y2 7xy2 4 4x2 2 3xy 2 10y2 20x2y 1 25xy2 33 2x2 1 3x 2x3 2 10x2 x2 2 8x 1 15 3x3 2 27x 4 14x 1 21 x2 2 6x 2 27 34 a2 2 4ab 1 4b2 6a2 2 4ab 3a2 1 5ab 2 2b2 6a2 1 ab 2 b2 4 a2 2 4b2 8a 1 4b 35 x 1 4 6 1 2x 2 1 4 36 3n 2 1 9 2 n 1 2 12 37 x 1 1 4 1 x 2 3 6 2 x 2 2 8 38 x 2 2 5 2 x 1 3 6 1 x 1 1 15 39 7 16a2b 1 3a 20b2 40 5b 24a2 2 11a 32b 41 1 n2 1 3 4n 2 5 6 42 3 n2 2 2 5n 1 4 3 43 3 4x 1 2 3y 2 1 44 5 6x 2 3 4y 1 2 45 3 2x 1 1 1 2 3x 1 4 46 5 x 2 1 2 3 2x 2 3 47 4x x2 1 7x 1 3 x 48 6 x2 1 8x 2 3 x 90360_ch0_001 102 indd 62 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied s
31. 0x0 c 02x0 2x d 0x0 2 02x0 In Problems 45 58 state the property that justi es each of the statements For example 3 1 24 5 24 1 3 because of the commutative property of addition Objective 5 45 x 2 5 2 x 46 7 1 4 1 6 5 7 1 4 1 6 47 1 x 5 x 48 43 1 218 5 218 1 43 49 21 93 5 293 50 109 1 2109 5 0 51 5 4 1 7 5 5 4 1 5 7 52 21 x 1 y 5 2 x 1 y 53 7yx 5 7xy 90360_ch0_001 102 indd 17 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 18 71 25x 1 8y 1 7y 1 8x x 5 5 and y 5 26 72 0 x 2 y0 2 0 x 1 y0 x 5 24 and y 5 27 73 0 3x 1 y0 1 0 2x 2 4y0 x 5 5 and y 5 23 74 x 2 y y 2 x x 5 26 and y 5 13 75 2a 2 3b 3b 2 2a a 5 24 and b 5 28 76 5 x 2 1 1 7 x 1 4 x 5 3 77 2 3x 1 4 2 3 2x 2 1 x 5 22 78 24 2x 2 1 2 5 3x 1 7 x 5 21 79 5 a 2 3 2 4 2a 1 1 2 2 a 2 4 a 5 23 80 23 2y 2 7 2 y 1 10 1 8y 1 5 y 5 10 For Problems 81
32. 1 6x 1 8 5 3 x 1 4 x 1 2 3x 3 2 3y 3 5 3 x 3 2 y 3 5 3 x 2 y x 2 1 xy 1 y 2 a 4 2 b 4 5 a 2 1 b 2 a 2 2 b 2 5 a 2 1 b 2 a 1 b a 2 b x 4 2 6x 2 2 27 5 x 2 2 9 x 2 1 3 5 x 1 3 x 2 3 x 2 1 3 3x 4y 1 9x 2y 2 84y 5 3y x 4 1 3x 2 2 28 5 3y x 2 1 7 x 2 2 4 5 3y x 2 1 7 x 1 2 x 2 2 x 2 2 y 2 1 8y 2 16 5 x 2 2 y 2 2 8y 1 16 5 x 2 2 y 2 4 2 5 x 2 y 2 4 x 1 y 2 4 5 x 2 y 1 4 x 1 y 2 4 For Problems 1 8 answer true or false 1 The process of expressing a polynomial as a product of polynomials is called factoring 2 x 2 5x 2 10 is the completely factored form of 5x 2 2 10x 2 3 The polynomial 3a3b 2 4c2d 1 5bd does not have a common factor 4 The sum of two squares is not factorable using integers 5 The sum of two cubes is not factorable using integers 6 A factoring problem can be partially checked by making sure the product of the factors equals the polynomial 7 All trinomials are factorable using integers 8 All common factors are monomial factors For Problems 1 6 factor completely by factoring out the common factor Objective 1 1 6xy 2 8xy 2 2 4a2b2 1 12ab3 3 12x2y3z4 2 6x4y3z3 1 6x2y3z2 4 3m2n
33. 1y2 2 x x 1 1 x2 1 2x 2 1y2 x2 1 2x 1y2 2 64 3x 1y3 2 x 3x 2 2y3 3x 1y3 2 65 3 2x 1y3 2 2x 2x 2 2y3 2x 1y3 2 66 Use your calculator to evaluate each of the following a 2 3 1728 b 2 3 5832 c 2 4 2401 d 2 4 65 536 e 2 5 161 051 f 2 5 6 436 343 67 In De nition 0 7 we stated that bmyn 5 2 n bm 5 A 2 n bBm Use your calculator to verify each of the following a 2 3 272 5 A 2 3 27B2 b 2 3 85 5 A 2 3 8B5 c 2 4 163 5 A 2 4 16B3 d 2 3 162 5 A 2 3 16B2 e 2 5 94 5 A 2 5 9B4 f 2 3 124 5 A 2 3 12B4 68 Use your calculator to evaluate each of the following a 165y2 b 257y2 c 169y4 d 275y3 e 3432y3 f 5124y3 69 Use your calculator to estimate each of the follow ing to the nearest thousandth a 74y3 b 104y5 c 122y5 d 192y5 e 73y4 f 105y4 Answers to the concept Quiz 1 C 2 B 3 A 4 B 5 True 6 True 7 True 8 True Graphing calculator Activities 90360_ch0_001 102 indd 81 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequ
34. 2 1 17x 1 5 solution 6x 2 1 17x 1 5 Sum of 17 Product of 6 5 5 30 classroom example Factor 2y2 1 11y 1 6 classroom example Factor 8a2 1 14a 1 3 90360_ch0_001 102 indd 46 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 4 Factoring Polynomials 47 We need two integers whose sum is 17 and whose product is 30 The integers 2 and 15 satisfy these conditions Therefore the middle term 17x of the given trinomial can be expressed as 2x 1 15x and we can proceed as follows 6x 2 1 17x 1 5 5 6x 2 1 2x 1 15x 1 5 5 2x 3x 1 1 1 5 3x 1 1 Factor by grouping 5 3x 1 1 2x 1 5 eXAMPLe 9 Factor 5x 2 2 18x 2 8 solution 5x 2 2 18x 2 8 Sum of 218 Product of 5 28 5 240 We need two integers whose sum is 218 and whose product is 240 The integers 220 and 2 satisfy these conditions Therefore the middle term 218x of the trinomial can be written 220x 2x and we can facto
35. 2 5 a 2 1 2ab 1 b 2 a 2 b 2 5 a 2 2 2ab 1 b 2 a 1 b a 2 b 5 a 2 2 b 2 a 1 b 3 5 a 3 1 3a 2b 1 3ab 2 1 b 3 a 2 b 3 5 a 3 2 3a 2b 1 3ab 2 2 b 3 The three following examples illustrate the rst three patterns respectively 2x 1 3 2 5 2x 2 1 2 2x 3 1 3 2 5 4x 2 1 12x 1 9 5x 2 2 2 5 5x 2 2 2 5x 2 1 2 2 5 25x 2 2 20x 1 4 3x 1 2y 3x 2 2y 5 3x 2 2 2y 2 5 9x 2 2 4y 2 In the rst two examples the resulting trinomial is called a perfect square trinomial it is the result of squaring a binomial In the third example the resulting binomial is called the difference of two squares Later we will use both of these patterns exten sively when factoring polynomials The cubing of a binomial patterns are helpful primarily when you are multiplying These patterns can shorten the work of cubing a binomial as the next two examples illustrate 3x 1 2 3 5 3x 3 1 3 3x 2 2 1 3 3x 2 2 1 2 3 5 27x 3 1 54x 2 1 36x 1 8 5x 2 2y 3 5 5x 3 2 3 5x 2 2y 1 3 5x 2y 2 2 2y 3 5 125x 3 2 150x 2y 1 60xy 2 2 8y 3 Keep in mind that these multiplying patterns are useful shortcuts but if you forget them simply revert to applying the distributive property 90360_ch0_001 102 indd 35 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due t
36. 2 20 x15y8 y 55 52 12 x9y8 4x 57 a 2 6 2 c 2x 61 2x 2 2 2x 2 1 32 63 x x2 1 2x 32 65 4x 2x 43 69 a 13 391 c 2 702 e 4 304 Problem Set 0 8 page 88 1 13 1 8i 3 3 1 4i 5 211 1 i 7 21 2 2i 9 2 3 20 1 5 12 i 11 7 10 2 11 12 i 13 4 1 0i 15 3i 17 i219 19 2 3 i 21 2i22 23 3i23 25 3i26 27 18i 29 12i22 31 22 2 i23 33 21 2 i22 35 4 1 i25 2 37 28 39 226 41 2225 43 22215 45 22214 47 3 49 26 51 221 1 0i 53 8 1 12i 55 0 1 26i 57 53 2 26i 59 10 2 24i 61 214 2 8i 63 27 1 24i 65 23 1 4i 67 113 1 0i 69 13 1 0i 71 2 8 13 1 12 13 i 73 1 2 2 3 i 75 0 2 3 2 i
37. 2 b 1 1 5 b 1 1 a 2 1 2 x 2y 2 1 2 y 2y 2 1 5 2y 2 1 x 2 y x x 1 2 1 3 x 1 2 5 x 1 2 x 1 3 Factoring by Grouping It may seem that a given polynomial exhibits no apparent common monomial or binomial factor Such is the case with ab 1 3c 1 bc 1 3a However by using the commutative property to rearrange the terms we can factor it as follows ab 1 3c 1 bc 1 3a 5 ab 1 3a 1 bc 1 3c 5 a b 1 3 1 c b 1 3 Factor a from the first two terms and c from the last two terms 5 b 1 3 a 1 c Factor b 1 3 from both terms This factoring process is referred to as factoring by grouping Let s consider another example of this type ab 2 2 4b 2 1 3a 2 12 5 b 2 a 2 4 1 3 a 2 4 Factor b2 from the first two terms 3 from the last two 5 a 2 4 b2 1 3 Factor the common binomial from both terms Difference of Two squares In Section 0 3 we called your attention to some special multiplication patterns One of these patterns was a 1 b a 2 b 5 a 2 2 b 2 90360_ch0_001 102 indd 42 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that
38. 2150a3b2 Assume all variables rep resent nonnegative values solution 2150a3b2 5 225a2b226a 5 5ab26a OBjecTiVe sUMMARy eXAMPLe continued Simplify an indicated sum of radical expressions section 0 6 Objective 3 The distributive property can be used to combine radicals that have the same index and the same radicand Sometimes the problem requires that the given radicals be ex pressed in simplest form Simplify 224 2 254 1 826 solution 224 2 254 1 826 5 2426 2 2926 1 826 5 226 2 326 1 826 5 726 Chapter 0 Summary 90360_ch0_001 102 indd 97 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 98 Multiply radical expressions section 0 6 Objective 4 Property 0 3 can be viewed as 2 n b2 n c 5 2 n bc This property along with commutative associ ative and distributive properties of real numbers provides a basis for multiplying radicals that have the same index Multiply 22xA 26x 1 218x
39. 2x 2 xy 1 2y 40 64x 3 2 27y 3 41 15x 2 2 14x 2 8 42 3x 3 1 36 43 2x 2 2 x 2 8 44 3x 3 1 24 45 x 4 2 13x 2 1 36 46 4x 2 2 4x 1 1 2 y 2 For Problems 47 56 perform the indicated operations involving rational expressions Express nal answers in simplest form 47 8xy 18x2y 24xy2 16y3 48 214a2b2 6b3 4 21a 15ab 49 x2 1 3x 2 4 x2 2 1 3x2 1 8x 1 5 x2 1 4x 50 9x2 2 6x 1 1 2x2 1 8 8x 1 20 6x2 1 13x 2 5 51 3x 2 2 4 1 5x 2 1 3 52 2x 2 6 5 2 x 1 4 3 53 3 n2 1 4 5n 2 2 n 54 5 x2 1 7x 2 3 x 55 3x x2 2 6x 2 40 1 4 x2 2 16 90360_ch0_001 102 indd 100 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 101 56 2 x 2 2 2 2 x 1 2 2 4 x3 2 4x For Problems 57 59 simplify each complex fraction 57 3 x 2 2 y 5 x2 1 7 y 58 3 2 2 x 4 1 3 x 59 3 x 1 h 2 2 3 x2 h 60 Simplify the expression 6 x2 1 2 1y2 2 6x2 x2 1 2 21y2
40. 2x is zero For example If x 5 4 then 2x 5 2 4 5 24 If x 5 22 then 2x 5 2 22 5 2 If x 5 0 then 2x 5 2 0 5 0 The concept of absolute value can be interpreted on the number line Geometrically the absolute value of any real number is the distance between that number and zero on the number line For example the absolute value of 2 is 2 the absolute value of 23 is 3 and the absolute value of zero is zero see Figure 0 6 2 1 0 1 2 3 3 0 0 3 3 2 2 Figure 0 6 Symbolically absolute value is denoted with vertical bars Thus we write 0 2 0 5 2 0 230 5 3 and 0 0 0 5 0 More formally the concept of absolute value is de ned as follows Definition 0 1 For all real numbers a 1 If a 0 then 0 a 0 5 a 2 If a 0 then 0 a 0 5 2a According to De nition 0 1 we obtain 0 6 0 5 6 by applying part 1 0 0 0 5 0 by applying part 1 0270 5 2 27 5 7 by applying part 2 Notice that the absolute value of a positive number is the number itself but the absolute value of a negative number is its opposite Thus the absolute value of any num ber except zero is positive and the absolute value of zero is zero Together these facts 90360_ch0_001 102 indd 7 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChap
41. 4 2 1 6 i Multiplying and Dividing complex Numbers Because i 2 5 21 the number i is a square root of 21 so we write i 5 121 It should also be evident that 2i is a square root of 21 because 2i 2 5 2i 2i 5 i 2 5 21 Therefore in the set of complex numbers 21 has two square roots namely i and 2i This is expressed symbolically as i 5 221 and 2i 5 2221 Let s extend the definition so that in the set of complex numbers every negative real number has two square roots For any positive real number b Ai2bB2 5 i2 b 5 21 b 5 2b Therefore let s denote the principal square root of 2b by 12b and define it to be 22b 5 i2b where b is any positive real number In other words the principal square root of any negative real number can be represented as the product of a real number and the imag inary unit i Consider the following examples 224 5 i24 5 2i 2217 5 i217 2224 5 i224 5 i2426 5 2i26 Note that we simplified the radical 124 to 216 We should also observe that 212b where b 0 is a square root of 2b because A222bB2 5 A2i2bB2 5 i 2 b 5 21 b 5 2b Thus in the set of complex numbers 2b where b 0 has two square roots i1b and 2i1b These are expressed as 22b 5 i2b and 222b 5 2i2b We must be careful with the use of the symbol 12b where b 0 Some proper ties that are true in the set of real numbers involving the square root symbol do not hold if the square root
42. 40 2ab2 3 a4b5 41 2 3 428 42 2 3 9227 58 Your friend keeps getting an error message when evaluating 245 2 on his calculator What error is he probably making 59 Explain how you would evaluate 27 2 gt 3 without a calculator Thoughts into Words 90360_ch0_001 102 indd 80 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 7 Relationship Between Exponents and Roots 81 Further investigations Sometimes we meet the following type of simplifica tion problem in calculus x 2 1 1y2 2 x x 2 1 2 1y2 x 2 1 1y2 2 5 a x 2 1 1y2 2 x x 2 1 2 1y2 x 2 1 2y2 b a x 2 1 1y2 x 2 1 1y2b 5 x 2 1 2 x x 2 1 0 x 2 1 3y2 5 x 2 1 2 x x 2 1 3y2 5 21 x 2 1 3y2 or 2 1 x 2 1 3y2 For Problems 60 65 simplify each expression as we did in the previous example 60 2 x 1 1 1y2 2 x x 1 1 2 1y2 x 1 1 1y2 2 61 2 2x 2 1 1y2 2 2x 2x 2 1 2 1y2 2x 2 1 1y2 2 62 2x 4x 1 1 1y2 2 2x2 4x 1 1 2 1y2 4x 1 1 1y2 2 63 x2 1 2x
43. 5 3 x 1 3 2 5 3x 1 6 3x x 1 4 5 3x x 1 3x 4 5 3x 2 1 12x For factoring purposes the distributive property now in the form ab 1 ac 5 a b 1 c can be used to reverse the process 3x 1 6 5 3 x 1 3 2 5 3 x 1 2 3x 2 1 12x 5 3x x 1 3x 4 5 3x x 1 4 Polynomials can be factored in a variety of ways Consider some factorizations of 3x 2 1 12x 3x2 1 12x 5 3x x 1 4 or 3x2 1 12x 5 3 x2 1 4x or 3x2 1 12x 5 x 3x 1 12 or 3x2 1 12x 5 1 2 6x2 1 24x We are however primarily interested in the first of these factorization forms we refer to it as the completely factored form A polynomial with integral coefficients is in completely factored form if 1 it is expressed as a product of polynomials with integral coefficients and 2 no polynomial other than a monomial within the factored form can be further factored into polynomials with integral coefficients Do you see why only the first of the factored forms of 3x 2 1 12x is said to be in com pletely factored form In each of the other three forms the polynomial inside the 0 4 Factoring Polynomials O B j e c T i V e s 1 Factor out a common factor 2 Factor by grouping 3 Factor the difference of two squares 4 Factor trinomials 5 Factor the sum or difference of two cubes 6 Apply more than one factoring technique 90360_ch0_001 102 indd 41 11 17 11 8 47 AM Copyrig
44. 77 22 41 2 7 41 i 79 21 1 2i 81 2 17 10 1 1 10 i 83 5 13 2 1 13 i 89 a 2 1 11i c 211 1 2i e 27 2 24i b 22 2 2i d 24 1 0i f 4 2 4i chapter 0 Review Problem Set page 100 1 1 125 2 2 1 81 3 16 9 4 1 9 5 28 6 3 2 7 21 2 8 1 6 9 4 10 28 11 12x2y 12 230x76 13 48 a16 14 27y35 x2 15 4y5 x5 16 8y x712 17 16x6 y6 18 2a3b1 3 19 4x 2 1 20 23x 1 8 21 12a 2 19 22 20x2 2 11x 2 42 23 212x2 1 17x 2 6 24 235x2 1 22x 2 3 25 x3 1 x2 2 19x 2 28 26 6x3 2 x2 1 10x 1 6 27 25x2 2 30x 1 9 28 9x2 1 42x 1 49 29 8x3 2 12x2 1 6x 2 1 30 27x3 1 135x2 1 225x 1 125 90360_ANS1_A1 A44 indd 3 11 17 11 11 19 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplica
45. Chapter 0 Some Basic Concepts of Algebra A Review 44 Factoring Trinomials Expressing a trinomial as the product of two binomials is one of the most common factoring techniques used in algebra As before to develop a factoring technique we first look at some multiplication ideas Let s consider the product x 1 a x 1 b using the distributive property to show how each term of the resulting trinomial is formed x 1 a x 1 b 5 x x 1 b 1 a x 1 b 5 x x 1 x b 1 a x 1 a b 5 x 2 1 a 1 b x 1 ab Notice that the coefficient of the middle term is the sum of a and b and that the last term is the product of a and b These two relationships can be used to factor trinomials Let s consider some examples eXAMPLe 1 Factor x 2 1 12x 1 20 solution We need two integers whose sum is 12 and whose product is 20 The numbers are 2 and 10 and we can complete the factoring as follows x 2 1 12x 1 20 5 x 1 2 x 1 10 eXAMPLe 2 Factor x 2 2 3x 2 54 solution We need two integers whose sum is 23 and whose product is 254 The integers are 29 and 6 and we can factor as follows x 2 2 3x 2 54 5 x 2 9 x 1 6 eXAMPLe 3 Factor x 2 1 7x 1 16 solution We need two integers whose sum is 7 and whose product is 16 The only possible pairs of factors of 16 are 1 16 2 8 and 4 4 A su
46. Objectives 2 and 3 19 4x2 5y2 15xy 24x2y2 20 5xy 8y2 18x2y 15 21 214xy4 18y2 24x2y3 35y2 22 6xy 9y4 30x3y 248x 23 7a2b 9ab3 4 3a4 2a2b2 24 9a2c 12bc2 4 21ab 14c3 concept Quiz 0 5 Problem set 0 5 90360_ch0_001 102 indd 61 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 62 49 4a 2 4 a2 2 4 2 3 a 1 2 50 6a 1 4 a2 2 1 2 5 a 2 1 51 3 x 2 1 2 2 4x 2 4 52 3x 1 2 4x 2 12 1 2x 6x 2 18 53 4 n2 2 1 1 2 3n 1 3 54 5 n2 2 4 2 7 3n 2 6 55 3 x 1 1 1 x 1 5 x2 2 1 2 3 x 2 1 56 5 x 2 5x 2 30 x2 1 6x 1 x x 1 6 57 5 x2 1 10x 1 21 1 4 x2 1 12x 1 27 58 8 a2 2 3a 2 18 2 10 a2 2 7a 2 30 59 5 x2 2 1 2 2 x2 1 6x 2 16 60 4 x2 1 2 2 7 x2 1 x 2 12 61 3x x2 2 6x 1 9 2 2 x 2 3 62 6 x2 2 9 2 9 x2 2 6x 1 9 63 x 2 x2 x 2 1 1 1 x2 2 1 64
47. a 1 21 73 2x 2n 1 7x n 2 30 74 3x 2n 2 16x n 2 12 75 x 4n 2 y 4n 76 16x 2a 1 24x a 1 9 77 Suppose that we want to factor x 2 1 34x 1 288 We need to complete the following with two num bers whose sum is 34 and whose product is 288 x 2 1 34x 1 288 5 x 1 __ x 1 __ These numbers can be found as follows Because we need a product of 288 let s consider the prime factorization of 288 288 5 25 32 Now we need to use five 2s and two 3s in the statement 1 5 34 Because 34 is divisible by 2 but not by 4 four factors of 2 must be in one number and one factor of 2 in the other number Also because 34 is not divisible by 3 both factors of 3 must be in the same number These facts aid us in determining that 2 2 2 2 1 2 3 3 5 34 or 16 1 18 5 34 Thus we can complete the original factoring prob lem x 2 1 34x 1 288 5 x 1 16 x 1 18 Use this approach to factor each of the following expressions a x 2 1 35x 1 96 b x 2 1 27x 1 176 c x 2 2 45x 1 504 d x 2 2 26x 1 168 e x 2 1 60x 1 896 f x 2 2 84x 1 1728 For Problems 19 36 factor each trinomial Indicate any that are not factorable using integers Objective 4 19 x 2 2 5x 2 14 20 a 2 1 5a 2 24 21 15 2 2x 2 x 2 22 40 2 6x
48. any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 4 Factoring Polynomials 43 This same pattern viewed as a factoring pattern a 2 2 b 2 5 a 1 b a 2 b is referred to as the difference of two squares Applying the pattern is a fairly simple process as these next examples illustrate x 2 2 16 5 x 2 2 4 2 5 x 1 4 x 2 4 4x 2 2 25 5 2x 2 2 5 2 5 2x 1 5 2x 2 5 Because multiplication is commutative the order in which we write the factors is not important For example x 1 4 x 2 4 can also be written x 2 4 x 1 4 You must be careful not to assume an analogous factoring pattern for the sum of two squares it does not exist For example x 2 1 4 x 1 2 x 1 2 because x 1 2 x 1 2 5 x 2 1 4x 1 4 We say that a polynomial such as x 2 1 4 is not fac torable using integers Sometimes the difference of two squares pattern can be applied more than once as the next example illustrates 16x 4 2 81y 4 5 4x 2 1 9y 2 4x 2 2 9y 2 5 4x 2 1 9y 2 2x 1 3y 2x 2 3y It may also happen that the squares are not just simple monomial squares These next three examples illustrate such polynomial
49. are not associative operations For example 8 2 6 2 10 5 28 but 8 2 6 2 10 5 12 An example showing that division is not associative is 8 4 4 4 2 5 1 but 8 4 4 4 2 5 4 Zero is the identity element for addition This means that the sum of any real number and zero is identically the same real number For example 287 1 0 5 0 1 287 5 287 One is the identity element for multiplication The product of any real number and 1 is identically the same real number For example 2119 1 5 1 2119 5 2119 90360_ch0_001 102 indd 9 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 10 The real number 2a is called the additive inverse of a or the opposite of a The sum of a number and its additive inverse is the identity element for addition For ex ample 16 and 216 are additive inverses and their sum is zero The additive inverse of zero is zero The real number 1 gt a is called the multiplicative inverse or reciprocal
50. finite sum of monomials 5 A polynomial with three terms is classified as a binomial 6 x 2 6 2 5 x2 1 36 7 A perfect square trinomial is the result when a trinomial is squared 8 Row number 4 in Pascal s triangle contains the coefficients of the expansion of a 1 b 3 For Problems 1 10 perform the indicated operations Objective 1 1 5x 2 2 7x 2 2 1 9x 2 1 8x 2 4 2 29x 2 1 8x 1 4 1 7x 2 2 5x 2 3 3 14x 2 2 x 2 1 2 15x 2 1 3x 1 8 4 23x 2 1 2x 1 4 2 4x 2 1 6x 2 5 5 3x 2 4 2 6x 1 3 1 9x 2 4 6 7a 2 2 2 8a 2 1 2 10a 2 2 7 8x 2 2 6x 2 2 1 x 2 2 x 2 1 2 3x 2 2 2x 1 4 8 12x2 1 7x 2 2 2 3x2 1 4x 1 5 1 24x2 2 7x 2 2 9 5 x 2 2 2 4 x 1 3 2 2 x 1 6 10 3 2x 2 1 2 2 3x 1 4 2 4 5x 2 1 concept Quiz 0 3 Problem set 0 3 90360_ch0_001 102 indd 38 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 3 Polynomials 39 For Problems 11 54 nd the indicated products Remember the special patterns that we discussed i
51. handling special products involving radicals as the next examples illustrate 222 A423 2 526B 5 A222BA423B 2 A222BA526B 5 826 2 10212 5 826 2 102423 5 826 2 2023 A222 2 27BA322 1 527B 5 222A322 1 527B 2 27A322 1 527B 5 A222BA322B 1 A222BA527B 2 A 27BA322B 2 A 27BA527B 5 6 2 1 10214 2 3214 2 5 7 5 223 1 7214 A 25 1 22BA 25 2 22B 5 25A 25 2 22B 1 22A 25 2 22B 5 A 25BA 25B 2 A 25BA 22B 1 A 22BA 25B 2 A 22BA 22B 5 5 2 210 1 210 2 2 5 3 Pay special attention to the last example It fits the special product pattern a 1 b a 2 b 5 a 2 2 b 2 We will use that idea in a moment More About simplest Radical Form Another property of nth roots is motivated by the following examples B 36 9 5 24 5 2 and 236 29 5 6 3 5 2 B 3 64 8 5 2 3 8 5 2 and 2 3 64 2 3 8 5 4 2 5 2 In general the following property can be stated Property 0 4 B n b c 5 2 n b 1 n c if 2 n b and 1 n c are real numbers and c 0 90360_ch0_001 102 indd 68 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if sub
52. how to add 2x 2 1 4 3x 1 5 14 96 Look back at the two approaches shown in Example 7 Which approach would you use to simplify 1 4 1 1 6 1 2 2 3 4 Which approach would you use to simplify Answers to the concept Quiz 1 True 2 False 3 False 4 True 5 False 6 True 7 True 8 If x 5 2 or x 5 22 97 Use the graphing feature of your graphing calcula tor to give visual support for your answers for Problems 60 68 98 For each of the following use your graphing calcu lator to help you decide whether the two given ex pressions are equivalent for all de ned values of x a 6x2 2 7x 1 2 8x2 1 6x 2 5 and 3x 2 2 4x 1 5 b 4x2 2 15x 2 54 4x2 1 13x 1 9 and x 2 6 x 1 1 c 2x2 1 3x 2 2 12x2 1 19x 1 5 and 2x 2 1 4x 1 5 d x3 1 2x2 2 3x x3 1 6x2 1 5x 2 12 and x x 1 4 e 25x2 2 11x 1 2 3x2 2 13x 1 14 and 25x 2 1 3x 2 7 0 6 Radicals O B j e c T i V e s 1 Evaluate roots of numbers 2 Write radical expressions in simplest radical form 3 Simplify an indicated sum of radical expressions 4 Multiply radical expressions 5 Rationalize radical expressions Recall from our work with exponents that to square a number means to raise it to the second power that is to use the number as a factor twice For example 42 5 4 4 5 16 and 24 2 5 24 24 5 16 A square root of a number is one of its two equal factors Graphing calculator Act
53. indd 50 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 51 82 Consider the following approach to factoring 12x 2 1 54x 1 60 12x 2 1 54x 1 60 5 3x 1 6 4x 1 10 5 3 x 1 2 2 2x 1 5 5 6 x 1 2 2x 1 5 Is this factoring process correct What can you suggest to the person who used this approach 78 Describe in words the pattern for factoring the sum of two cubes 79 What does it mean to say that the polynomial x 2 1 5x 1 7 is not factorable using integers 80 What role does the distributive property play in the factoring of polynomials 81 Explain your thought process when factoring 30x 2 1 13x 2 56 Thoughts into Words Answers to the concept Quiz 1 True 2 False 3 True 4 True 5 False 6 True 7 False 8 False 0 5 Rational Expressions O B j e c T i V e s 1 Simplify rational expressions 2 Multiply and divide rational expressions 3 Add and subtract r
54. m 5 bnm 5 a1 5 a 16y2y3 1y2 5 16 1y2 y2y3 1y2 ab n 5 anbn 5 4y1y3 y3y4 y1y2 5 y3y421y2 bn bm 5 bn2m 5 y3y422y4 5 y1y4 ax1y2 y1y3b 6 5 x1y2 6 y1y3 6 aa bb n 5 an bn 5 x3 y2 The link between exponents and roots provides a basis for multiplying and dividing some radicals even if they have different indexes The general procedure is to change from radical to exponential form apply the properties of exponents and then change back to radical form Let s apply these procedures in the next three examples 222 3 2 5 21y2 21y3 5 21y211y3 5 25y6 5 2 6 25 5 2 6 32 2xy2 5 x2y 5 xy 1y2 x2y 1y5 5 x1y2y1y2x2y5y1y5 5 x1y212y5y1y211y5 5 x9y10y7y10 5 x9y7 1y10 5 2 10 x9y7 25 2 3 5 5 51y2 51y3 5 51y221y3 5 51y6 5 2 6 5 90360_ch0_001 102 indd 77 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 78 Earlier we agreed that a radical such as 2 3 x4 is not
55. may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 6 Radicals 67 In general the following property can be stated Property 0 3 2 n bc 5 2 n b2 n c if 2 n b and 2 n c are real numbers Property 0 3 states that the nth root of a product is equal to the product of the nth roots The definition of nth root along with Property 0 3 provides the basis for changing radicals to simplest radical form The concept of simplest radical form takes on ad ditional meaning as we encounter more complicated expressions but for now it simply means that the radicand does not contain any perfect powers of the index Consider the following examples of reductions to simplest radical form 245 5 29 5 5 2925 5 325 252 5 24 13 5 24213 5 2213 2 3 24 5 2 3 8 3 5 2 3 82 3 3 5 22 3 3 A variation of the technique for changing radicals with index n to simplest form is to factor the radicand into primes and then to look for the perfect nth powers in expo nential form as in the following examples 280 5 224 5 5 22425 5 2225 5 425 2 3 108 5 2 3 22 33 5 2 3 332 3 22 5 32 3 4 The distributive property can be used to combine radicals that have the same in
56. negative if the original number is less than 1 b positive if the original number is greater than 10 and c 0 if the original number itself is between 1 and 10 Thus we can write 0 00092 5 9 2 1024 872 000 000 5 8 72 108 5 1217 5 5 1217 100 To change from scienti c notation to ordinary decimal notation the following proce dure can be used Move the decimal point the number of places indicated by the exponent of 10 Move the decimal point to the right if the exponent is positive Move it to the left if the exponent is negative Thus we can write 3 14 107 5 31 400 000 7 8 1026 5 0 0000078 Scienti c notation can be used to simplify numerical calculations We merely change the numbers to scienti c notation and use the appropriate properties of expo nents Consider the following examples 90360_ch0_001 102 indd 26 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 2 Exponents 27 eXAMPLe 10 Use scientific notation to perform the indicated operations
57. previously listed conditions However for certain problems in calculus it is necessary to rationalize the numerator Again the fact that A1a 1 2bBA1a 2 2bB 5 a 2 b can be used eXAMPLe 5 Change the form of 2x 1 h 2 2x h by rationalizing the numerator solution 2x 1 h 2 2x h 5 a 2x 1 h 2 2x h ba 2x 1 h 1 2x 2x 1 h 1 2x b 5 x 1 h 2 x hA2x 1 h 1 2xB 5 h hA2x 1 h 1 2xB 5 1 2x 1 h 1 2x Radicals containing Variables Before we illustrate how to simplify radicals that contain variables there is one im portant point we should call to your attention Let s look at some examples to illus trate the idea classroom example Simplify 7 25 2 23 by rationalizing the denominator classroom example Change the form of 23x 1 3h 2 23x h by rationalizing the numerator 90360_ch0_001 102 indd 71 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 72 Consider the radical 2x2 fo
58. review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 54 To divide rational expressions we merely apply the following property a b 4 c d 5 a b d c 5 ad bc That is the quotient of two rational expressions is the product of the first expression times the reciprocal of the second Consider the following examples x2 16x2y 24xy3 4 9xy 8x2y2 5 16x2y 24xy3 8x2y2 9xy 5 16 8 x4 y3 24 3 9 x2 y4 5 16x2 27y y 3a2 1 12 3a2 2 15a 4 a4 2 16 a2 2 3a 2 10 5 3a2 1 12 3a2 2 15a a2 2 3a 2 10 a4 2 16 5 3 a2 1 4 a 2 5 a 1 2 3a a 2 5 a2 1 4 a 1 2 a 2 2 5 1 a a 2 2 Adding and subtracting Rational expressions The following two properties provide the basis for adding and subtracting rational expressions a b 1 c b 5 a 1 c b a b 2 c b 5 a 2 c b These properties state that rational expressions with a common denominator can be added or subtracted by adding or subtracting the numerators and placing the result over the common denominator Let s illustrate this idea 8 x 2 2 1 3 x 2 2 5 8 1 3 x 2 2 5 11 x 2 2 9 4y 2 7 4y 5 9 2 7 4y 5 2 4y 5 1 2y Don
59. symbol does not represent a real number For example 1a1b 5 1ab does not hold if a and b are both negative numbers Correct 224229 5 2i 3i 5 6i2 5 6 21 5 26 Incorrect 224229 5 2 24 29 5 236 5 6 90360_ch0_001 102 indd 84 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 8 Complex Numbers 85 To avoid difficulty with this idea you should rewrite all expressions of the form 12b where b 0 in the form i1b before doing any computations The following examples further illustrate this point 225227 5 Ai25BAi27B 5 i2235 5 21 235 5 2235 222228 5 Ai22BAi28B 5 i2216 5 21 4 5 24 22228 5 Ai22BA 28B 5 i216 5 4i 226228 5 Ai26BAi28B 5 i2248 5 i221623 5 4i223 5 2423 222 23 5 i22 23 5 i22 23 23 23 5 i26 3 2248 212 5 i248 212 5 iB 48 12 5 i24 5 2i Because complex numbers have a binomial form we can find the product of two complex numbers in the same way that we find the product of two binomials Then by replacing i 2 w
60. the exponents of a and b is n Next let s arrange the coef cients in a triangular formation this yields an easy to remember pattern 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Row number n in the formation contains the coef cients of the expansion of a 1 b n For example the fth row contains 1 5 10 10 5 1 and these numbers are the coef cients of the terms in the expansion of a 1 b 5 Furthermore each can be formed from the previous row as follows 1 Start and end each row with 1 2 All other entries result from adding the two numbers in the row immediately above one number to the left and one number to the right Thus from row 5 we can form row 6 Row 5 1 5 10 10 5 1 Add Add Add Add Add Row 6 1 6 15 20 15 6 1 90360_ch0_001 102 indd 36 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 3 Polynomials 37 Now we can use these seven coef cients and our discussion about the exponents to write out the expansio
61. to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 100 Chapter 0 Review Problem set For Problems 1 10 evaluate 1 523 2 2324 3 a3 4b 22 4 1 a1 3b 22 5 2264 6 B 3 27 8 7 B 5 2 1 32 8 3621 2 9 a1 8b 22y3 10 2323 5 For Problems 11 18 perform the indicated operations and simplify Express the nal answers using positive exponents only 11 3x 22y 21 4x 4y 2 12 5x 2y3 26x 1y2 13 28a 21y2 26a 1y3 14 3x 22y3y 1y5 3 15 64x22y3 16x3y22 16 56x21y3y2y5 7x1y4y23y5 17 a28x2y21 2x21y2 b 2 18 a 36a21b4 212a2b5b 21 For Problems 19 34 perform the indicated operations 19 27x 2 3 1 5x 2 2 1 6x 1 4 20 12x 1 5 2 7x 2 4 2 8x 1 1 21 3 a 2 2 2 2 3a 1 5 1 3 5a 2 1 22 4x 2 7 5x 1 6 23 23x 1 2 4x 2 3 24 7x 2 3 25x 1 1 25 x 1 4 x2 2 3x 2 7 26 2x 1 1 3x 2 2 2x 1 6 27 5x 2 3 2 28 3x 1 7 2 29 2x 2 1 3 30 3x 1 5 3 31 x 2 2 2x 2 3 x 2 1 4x 1 5 32 2x 2 2 x 2 2 x 2 1 6x 2 4 33 24x3y4 2 48x2y3 26xy 34 256x2y 1 72x3y2 8x2 For Problems 35 46 factor each polynomial completely Indicate any that are not factorable using integers 35 9x 2 2 4y 2 36 3x 3 2 9x 2 2 120x 37 4x 2 1 20x 1 25 38 x 2 y 2 2 9 39 x 2 2
62. you do not have access to a graphing utility The examples are chosen to reinforce concepts under discussion Furthermore for those who do have access to a graphing utility we provide Graphing Calculator Activities in many of the problem sets Figure 0 12 Graphing calculators have display windows large enough to show graphs This window feature is also helpful when you re using a graphing calculator for computa tional purposes because it allows you to see the entries of the problem Figure 0 13 shows a display window for an example of the distributive property Note that we can check to see that the correct numbers and operational symbols have been entered Also note that the answer is given below and to the right of the problem Figure 0 13 Most calculators including graphing calculators can be used to evaluate algebraic expressions One calculator method for evaluating the algebraic expression in Example 1 3xy 2 4z for x 5 2 y 5 24 and z 5 25 is to replace x with 2 y with 24 and z with 25 and then calculate the resulting numerical expression Courtesy of Texas Instruments 90360_ch0_001 102 indd 15 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the o
63. 0 5 2 4 81 5 3 because 34 5 81 2 5 32 5 2 because 25 5 32 2 5 232 5 22 because 22 5 5 232 To complete our terminology the n in the radical 2 n b is called the index of the radical If n 5 2 we commonly write 2b instead of 2 2 b In this text when we use symbols such as 2 n b 1 m y and 1 r x we will assume the previous agreements relative to the existence of real roots without listing the various restrictions unless a special re striction is needed From Definition 0 5 we see that if n is any positive integer greater than 1 and 2 n b exists then A 2 n bBn 5 b For example A 24B2 5 4 A 2 3 28B3 5 28 and A 2 4 81B4 5 81 Furthermore if b 0 and n is any positive integer greater than 1 or if b 0 and n is an odd positive integer greater than 1 then 2 n bn 5 b For example 242 5 4 2 3 22 3 5 22 and 2 5 65 5 6 But we must be careful because 2 22 2 22 and 2 4 22 4 22 simplest Radical Form Let s use some examples to motivate another useful property of radicals 216 25 5 2400 5 20 and 216 225 5 4 5 5 20 2 3 8 27 5 2 3 216 5 6 and 2 3 8 2 3 27 5 2 3 5 6 2 3 28 64 5 2 3 2512 5 28 and 2 3 28 2 3 64 5 22 4 5 28 90360_ch0_001 102 indd 66 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content
64. 00 We can proceed as follows 2640 000 5 2 64 10 4 5 64 10 4 1y2 5 64 1y2 104 1y2 5 8 10 2 5 8 100 5 800 Compute each of the following without a calculator and then use a calculator to check your answers a 249 000 000 b 20 0025 c 214 400 d 20 000121 e 2 3 27 000 f 2 3 0 000064 102 There are several methods of approximating square roots without using a calculator One such method works on a clamping between values principle For example to find a whole number approximation for 1128 we can proceed as follows 112 121 and 122 144 Therefore 11 1128 12 Because 128 is closer to 121 than to 144 we say that 11 is a whole number approximation for 1128 If a more precise approximation is needed we can do more clamping We would find that 11 3 2 127 69 and 11 4 2 129 96 Because 128 is closer to 127 69 than to 129 96 we conclude that 1128 11 3 to the nearest tenth For each of the following use the clamping idea to find a whole number approximation Then check your answers using a calculator and the square root key a 252 b 293 c 2174 d 2200 e 2275 f 2350 103 The clamping process discussed in Problem 102 works for any whole number root greater than or equal to 2 For example a whole number ap proximation for 2 3 80 is 4 because 43 64 and 53 125 and 80 is closer to 64 than to 125 For each of the following use the clamping idea to find a wh
65. 00001 37 17 72 39 1 6 41 48 19 43 1 x4 45 1 a2 47 1 a6 49 y4 x3 51 c3 a3b6 53 y2 4x4 55 x4 y6 57 9a2 4b4 59 1 x3 61 a3 b 63 220x4y5 65 227x3y9 67 8x6 27y9 69 28x6 71 6 x3y 73 6 a2y3 75 4x3 y5 77 2 5 a2b 79 1 4x2y4 81 x 1 1 x2 A 1 Answers to Odd Numbered Problems and All Chapter Review Chapter Test and Cumulative Review Problems chapter 0 90360_ANS1_A1 A44 indd 1 11 17 11 11 19 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Answers to Odd Numbered Problems A
66. 0016 400 b 24 000 000 90360_ch0_001 102 indd 27 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 28 A MODE key is often used on calculators to let you choose normal decimal nota tion scienti c notation or engineering notation The abbreviations Norm Sci and Eng are commonly used If the calculator is in scienti c mode then a number can be entered and changed to scienti c form with the ENTER key For example when we enter 589 and press the ENTER key the display will show 5 89E2 Likewise when the calculator is in scienti c mode the answers to computational problems are given in scienti c form For example the answer for 76 533 is given as 4 0508E4 It should be evident from this brief discussion that even when you are using a cal culator you need to have a thorough understanding of scienti c notation For Problems 1 10 answer true or false 1 Exponents are used to indicate repeated multiplications
67. 1 1y B 2 3 y2 y C 21 3 y 3 2w21 2 A 21w w B 1w C 21w 4 1 n x1 m x A 1 mn x B x m1n mn C x 1 m1n For Problems 5 8 answer true or false 5 Assuming the nth root of x exists 2 n x can be expressed as x 1 n 6 The expression 2 n xm is A1 n xBm 7 The process of rationalizing the denominator can be done with rational exponents 8 An exponent of 1 3 indicates the cube root classroom example Simplify 2 4 1 3 5 For Problems 1 16 evaluate Objective 1 1 491 gt 2 2 641 gt 3 3 323 gt 5 4 28 1 gt 3 5 282 gt 3 6 6421 gt 2 7 a1 4b 2 1y2 8 a2 27 8 b 2 1y3 9 163 gt 2 10 0 008 1 gt 3 11 0 01 3 gt 2 12 a 1 27b 2 2y3 13 6425 gt 6 14 2165 gt 4 15 a1 8b 2 1y3 16 a2 1 8b 2y3 Problem set 0 7 concept Quiz 0 7 90360_ch0_001 102 indd 79 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review
68. 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 97 Simplify complex fractions section 0 5 Objective 4 Fractions that contain rational expressions in the numerators or denominators are called complex fractions In Section 0 5 two methods were shown for simpli fying complex fractions Simplify 2 x 2 3 y 4 x2 1 5 y solution 2 x 2 3 y 4 x2 1 5 y Multiply the numerator and denominator by x2y x2ya2 x 2 3 yb x2ya 4 x2 1 5 yb 5 x2ya2 xb 1 x2ya23 yb x2ya 4 x2b 1 x2ya5 yb 5 2xy 2 3x2 4y 1 5x2 Write radical expressions in simplest radical form section 0 6 Objective 2 A radical expression is in sim plest form if 1 No fraction appears within a radical sign 2 No radical appears in the de nominator 3 No radicand contains a per fect power of the index The following properties are used to express radicals in sim plest form 2 n bc 5 2 n b2 n c B n b c 5 2 n b 2 n c Simplify
69. 2 45 3a 2 7 2 47 2n n2 1 3n 1 5 49 2n n2 1 7n 2 10 51 4 x 1 2 x2 2 2x 1 4 53 x 1 3 x 2 3 x2 1 5 55 2y x 1 4 x 2 4 x2 1 3 57 a 1 b 1 c 1 d a 1 b 2 c 2 d 59 x 1 4 1 y x 1 4 2 y 61 x 1 y 1 5 x 2 y 2 5 63 10x 1 3 6x 2 5 65 3x 7x 2 4 4x 1 5 67 xa 1 4 xa 2 4 69 xn 2 yn x2n 1 xnyn 1 y2n 71 xa 1 4 xa 2 7 73 2xn 2 5 xn 1 6 75 x2n 1 y2n xn 1 yn xn 2 yn 77 a x 1 32 x 1 3 c x 2 21 x 2 24 e x 1 28 x 1 32 Problem Set 0 5 page 61 1 2x 3 3 7y3 9x 5 8x4y4 9 7 a 1 4 a 2 9 9 x 2x 1 7 y x 1 9 11 x2 1 xy 1 y2 x 1 2y 13 2 2 x 1 1 15 2x 1 y x 2 y 17 9x2 2 6xy 1 4y2 x 2 5 19 x 2y3 21 28x3y3 15 23 14 27a 25 5y 27 5 a 1 3 a a 2 2 29 x 1 6y 2 2x 1 3y y3 x 1 4y 31 3xy 4 x 1 6 33 x 2 9 42x2 35 8x 1 5 12
70. 2 6m4n3 2 9m5n4 5 x z 1 3 1 y z 1 3 6 5 x 1 y 1 a x 1 y For Problems 7 10 factor completely by using group ing Objective 2 7 3x 1 3y 1 ax 1 ay 8 ac 1 bc 1 a 1 b 9 ax 2 ay 2 bx 1 by 10 2a2 2 3bc 2 2ab 1 3ac For Problems 11 18 factor by applying the difference of squares pattern Objective 3 11 9x 2 2 25 12 36x 2 2 121 13 1 2 81n 2 14 9x 2y 2 2 64 15 x 1 4 2 2 y 2 16 x 2 2 y 2 1 2 17 9s 2 2 2t 2 1 2 18 4a 2 2 3b 1 1 2 concept Quiz 0 4 Problem set 0 4 90360_ch0_001 102 indd 49 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 50 For Problems 67 76 factor each of the following and assume that all variables appearing as exponents rep resent integers 67 x 2a 2 16 68 x 4n 2 9 69 x 3n 2 y 3n 70 x 3a 1 y 6a 71 x 2a 2 3x a 2 28 72 x 2a 1 10x
71. 2 An exponent cannot be zero 3 222 5 24 4 21 22 5 2 5 In the expression 63 the number 6 is referred to as the baseline number 6 2 1 5 2 5 4 1 25 7 322 1 324 5 326 8 When writing a number in scientific notation n 10k the number n must be greater than 1 and less than or equal to 10 9 Single digit numbers can be expressed in scientific notation 10 The number 357 000 is written as 35 7 104 in scientific notation For Problems 1 42 evaluate each numerical expres sion Objective 1 1 223 2 322 3 21023 4 1024 5 1 323 6 1 225 7 a1 2b 22 8 2a1 3b 22 9 a22 3b 23 10 a5 6b 22 11 a21 5b 0 12 1 a3 5b 22 13 1 a4 5b 22 14 a4 5b 0 15 25 223 16 322 35 17 1026 104 18 106 1029 19 1022 1023 20 1021 1025 21 322 22 22 22 21 23 concept Quiz 0 2 Problem set 0 2 90360_ch0_001 102 indd 28 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBra
72. 2 2 1 5 2 2 x 5 8 x 2 2 2 5 x 2 2 5 8 2 5 x 2 2 5 3 x 2 2 Use this approach to do the following problems a 7 x 2 1 1 2 1 2 x b 5 2x 2 1 1 8 1 2 2x c 4 a 2 3 2 1 3 2 a d 10 a 2 9 2 5 9 2 a e x2 x 2 1 2 2x 2 3 1 2 x f x2 x 2 4 2 3x 2 28 4 2 x For Problems 70 92 simplify each complex fraction Objective 4 70 2 x 1 7 y 3 x 2 10 y 71 5 x2 2 3 x 1 y 1 2 y2 72 1 x 1 3 2 y 1 4 73 1 1 1 x 1 2 1 x 94 Explain in your own words how to multiply two rational expressions 93 What role does factoring play in the simplifying of rational expressions Thoughts into Words 90360_ch0_001 102 indd 63 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 64 5 8 1 4 9 5 14 2 2 21 Explain the reason for your choice of approach for each problem 95 Give a step by step description of
73. 2 23 21 2 3 48x4y5 For Problems 22 25 perform the indicated operations and express the resulting complex numbers in standard form 22 22 2 4i 2 21 1 6i 1 23 1 7i 23 5 2 7i 4 1 2i 24 7 2 6i 7 1 6i 25 1 1 2i 3 2 i Chapter 0 Test 102 90360_ch0_001 102 indd 102 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Unless otherwise noted all art on this page is Cengage Learning Problem Set 0 1 page 16 1 True 3 False 5 False 7 True 9 False 11 5466 13 50 214 466 15 5 15 212 2p6 17 50 2146 19 21 23 25 27 29 31 33 516 35 50 1 2 36 37 5 22 21 0
74. 20 5 0 Zero has only one square root Technically we could also write 220 5 20 5 0 224 Not a real number 2224 Not a real number To cube a number means to raise it to the third power that is to use the number as a factor three times For example 23 5 2 2 2 5 8 and 22 3 5 22 22 22 5 28 A cube root of a number is one of its three equal factors Thus 2 is a cube root of 8 and as we will discuss later it is the only real number that is a cube root of 8 Furthermore 22 is the only real number that is a cube root of 28 In general a is a cube root of b if a3 5 b The following statements generalize these ideas 1 Every positive real number has one positive real number cube root 2 Every negative real number has one negative real number cube root 3 The cube root of zero is zero Remark Every nonzero real number has three cube roots but only one of them is a real number The other roots are complex numbers which we will discuss in Section 0 8 The symbol 1 3 is used to designate the cube root of a number Thus we can write 2 3 8 5 2 2 3 28 5 22 B 3 1 27 5 1 3 B 3 2 1 27 5 21 3 The concept of root can be extended to fourth roots fifth roots sixth roots and in general nth roots If n is an even positive integer then the following statements are true 1 Every positive real number has exactly two real nth roots one positive and one negative For example the re
75. 24 2 12 85 216 96 Explain the difference between simplifying a nu merical expression and evaluating an algebraic expression Thoughts into Words 90360_ch0_001 102 indd 18 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 2 Exponents 19 Positive integers are used as exponents to indicate repeated multiplication For exam ple 4 4 4 can be written 43 where the raised 3 indicates that 4 is to be used as a factor three times The following general de nition is helpful Definition 0 2 If n is a positive integer and b is any real number then bn 5 bbb p b 14243 n factors of b The number b is referred to as the base and n is called the exponent The expression b n can be read b to the nth power The terms squared and cubed are commonly as sociated with exponents of 2 and 3 respectively For example b2 is read b squared and b3 as b cubed An exponent of 1 is usually not written so b1 is simply written b The following
76. 2n 1 1 x3 1 y3 x2 1 xy 1 2x 1 2y 5 x 1 y x2 2 xy 1 y2 x x 1 y 1 2 x 1 y 5 x 1 y x2 2 xy 1 y2 x 1 y x 1 2 5 x2 2 xy 1 y2 x 1 2 6x3y 2 6xy x3 1 5x2 1 4x 5 6xy x2 2 1 x x2 1 5x 1 4 5 6xy x 1 1 x 2 1 x x 1 1 x 1 4 5 6y x 2 1 x 1 4 x 5 y 2 90360_ch0_001 102 indd 52 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 53 Note that in the last example we left the numerator of the final fraction in factored form This is often done if expressions other than monomials are involved Either 6y x 2 1 x 1 4 or 6xy 2 6y x 1 4 is an acceptable answer Remember that the quotient of any nonzero real number and its opposite is 21 For example 6 gt 26 5 21 and 28 gt 8 5 21 Likewise the indicated quotient of any poly nomial and its opposite is equal to 21 For example a 2a 5 21 because a and 2a are opposites a 2 b b 2 a 5 21 because a 2 b and b 2 a are opposites x2 2 4 4 2 x2 5 21 because x2 2 4 an
77. 2x2 2 3 8x3 5 32 3 2x2 2x For Problems 1 8 answer true or false 1 The symbol 1 is used to designate the principal square root 2 Every positive real number has two principal square roots 3 The square root of zero does not exist in the real number system 4 Every real number has one real number cube root 5 The 125 could be 5 or 25 concept Quiz 0 6 90360_ch0_001 102 indd 72 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 6 Radicals 73 For Problems 1 8 evaluate Objective 1 1 281 2 2249 3 2 3 125 4 2 4 81 5 B 36 49 6 B 256 64 7 B 3 227 8 8 B 3 64 27 For Problems 9 30 express each in simplest radical form All variables represent positive real numbers Objective 2 9 224 10 254 11 2112 12 6228 13 23244 14 25268 15 3 4220 16 3 8272 17 212x2 18 245xy2 19 264x4y7 20 3232a3 21 3 7245xy6 22 2 3 32 23 2 3 128 24 2 3 54x3 25 2 3 16x4 26 2 3 81x5y6 27
78. 2y3 y classroom example Perform the indicated operations and express the answers in simplest radical form a 2 3 x22x b 2 4 32 3 9 c 2 3 4 22 classroom example Rationalize the denominator and express the answer in simplest radical form a 3 2 3 y2 b 2m 2 5 n2 90360_ch0_001 102 indd 78 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 7 Relationship Between Exponents and Roots 79 Note in part b that if we had changed back to radical form at the step x1y3y1y2 y we would have obtained the product of two radicals 1 3x1y in the numerator Instead we used the exponential form to find this product and express the final result with a single radi cal in the numerator Finally let s consider an example involving the root of a root eXAMPLe 3 Simplify 2 3 12 solution 2 3 12 5 A21y2B1y3 5 21y6 5 2 6 2 For Problems 1 4 select the equivalent radical form 1 x 3 5 A 2 3 x5 B x2 3 x2 C 2 5 x3 2 y21 3 A
79. 2y6 1 y7 65 32a5 2 240a4b 1 720a3b2 2 1080a2b3 1 810ab4 2 243b5 67 3x2 2 5x 69 25a4 1 4a2 2 9a 71 5ab 1 11a2b4 73 x2a 2 y2b 75 x2b 2 3xb 2 28 77 6x2b 1 xb 2 2 79 x4a 2 2x2a 1 1 81 x3a 2 6x2a 1 12xa 2 8 Problem Set 0 4 page 49 1 2xy 3 2 4y 3 6x2y3z2 2z2 2 x2z 1 1 5 z 1 3 x 1 y 7 x 1 y 3 1 a 9 x 2 y a 2 b 11 x 1 5 x 2 5 13 1 1 9n 1 2 9n 15 x 1 4 1 y x 1 4 2 y 17 3s 1 2t 2 1 3s 2 2t 1 1 19 x 2 7 x 1 2 21 5 1 x 3 2 x 23 Not factorable 25 3x 2 5 x 2 2 27 10x 1 7 x 1 1 29 5x 2 3 2x 1 9 31 6a 2 1 2 33 4x 2 y 2x 1 y 35 Not factorable 37 x 2 2 x2 1 2x 1 4 39 4x 1 3y 16x2 2 12xy 1 9y2 41 4 x2 1 4 43 x x 1 3 x 2 3 90360_ANS1_A1 A44 indd 2 11 17 11 11 19 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be sup
80. 3 108 0 000075 4 800 000 15 000 0 0012 109 2900 000 000 110 20 000004 111 20 0009 112 0 00069 0 0034 0 0000017 0 023 For Problems 89 98 nd the following products and quotients Assume that all variables appearing as expo nents represent integers Objective 2 For example x 2b x 2b11 5 x 2b1 2b11 5 x b11 89 3x a 4x 2a11 90 5x 2a 26x 3a21 91 x a x 2a 92 22y 3b 24y b11 93 x3a xa 94 4x2a11 2xa22 95 224y5b11 6y2b21 96 xa 2b xb a 97 xy b yb 98 2x2b 24xb11 8x2b12 For Problems 99 102 express each number in scienti c notation Objective 3 99 62 000 000 100 17 000 000 000 101 0 000412 102 0 000000078 113 Explain how you would simplify 321 222 21 and also how you would simplify 321 1 222 21 114 How would you explain why the product of x 2 and x 4 is x 6 and not x 8 115 Use your calculator to check your answers for Problems 107 112 116 Use your calculator to evaluate each of the follow ing Express nal answers in ordinary notation a 27 000 2 b 450 000 2 c 14 800 2 d 1700 3 e 900 4 f 60 5 g 0 0213 2 h 0 000213 2 i 0 000198 2 j 0 000009 3 117 Use your calculator to estimate each of the follow ing Express nal answers in scienti c notation with the number between 1 and 10 rounded to the nearest one thousandth a 4576 4 b 719 10 c 28 12
81. 40 322 2 23 41 224 1 321 21 42 322 2 521 21 Simplify Problems 43 62 express nal results with out using zero or negative integers as exponents Objective 2 43 x 3 x 27 44 x 22 x 23 45 a 2 a 23 a 21 46 b 23 b 5 b 24 47 a 23 2 48 b 5 22 49 x 3y 24 21 50 x 4y 22 22 51 ab 2c 21 23 52 a 2b 21c 22 24 53 2x 2y 21 22 54 3x 4y 22 21 55 ax22 y23b 22 56 a y4 x21b 23 57 a2a21 3b22b 22 58 a 3x2y 4a21b23b 21 90360_ch0_001 102 indd 29 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 30 For Problems 103 106 change each number from scienti c notation to ordinary decimal form Objective 4 103 1 8 105 104 5 41 107 105 2 3 1026 106 4 13 1029 For Problems 107 112 use scienti c notation and the properties of exponents to help perform the indicated operations Objective 5 107 0 00052 0 01
82. 51yB For Problems 69 80 rationalize the denominator and simplify All variables represent positive real numbers Objective 5 69 3 25 1 2 70 7 210 2 3 88 Why is 129 not a real number 89 How could you find a whole number approxima tion for 12750 if you did not have a calculator or table available 85 Is the equation 2x2y 5 x1y true for all real number values for x and y Defend your answer 86 Is the equation 2x2y2 5 xy true for all real number values for x and y Defend your answer 87 Give a step by step description of how you would change 1252 to simplest radical form Further investigations Thoughts into Words 90360_ch0_001 102 indd 74 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 7 Relationship Between Exponents and Roots 75 101 Sometimes it is more convenient to express a large or very small number as a product of a power of 10 and a number that is not between 1 and 10 For example suppose that we want to cal culate 2640 0
83. 8 5 n 2 8 n 1 1 f LCD 5 n 1 1 n 1 5 n 2 8 classroom example Add 4m m2 1 3m 2 4 1 5 m2 1 m 2 2 90360_ch0_001 102 indd 57 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 58 3n n2 1 6n 1 5 1 4 n2 2 7n 2 8 5 c 3n n 1 5 n 1 1 d an 2 8 n 2 8b 1 c 4 n 2 8 n 1 1 d an 1 5 n 1 5b 5 3n n 2 8 n 1 5 n 1 1 n 2 8 1 4 n 1 5 n 1 5 n 1 1 n 2 8 5 3n2 2 24n 1 4n 1 20 n 1 5 n 1 1 n 2 8 5 3n2 2 20n 1 20 n 1 5 n 1 1 n 2 8 simplifying complex Fractions Fractional forms that contain rational expressions in the numerator and or the denomi nator are called complex fractions The following examples illustrate some approaches to simplifying complex fractions eXAMPLe 7 Simplify 3 x 1 2 y 5 x 2 6 y2 solution A Treating the numerator as the sum of two rational expressions and the denominator as the difference of two rational expressions we can proce
84. 80 43 22 2 3 2 44 29 2 3 3 45 2 3 8 2 4 4 46 2 3 16 2 6 4 47 2 4 x9 2 3 x2 48 2 5 x7 2 3 x For Problems 49 56 rationalize the denominator and express the final answer in simplest radical form Objective 3 49 5 2 3 x 50 3 2 3 x2 51 2x 2 3 y 52 2 4 x 2y 53 2 4 x3 2 5 y3 54 22x 32 3 y 55 52 3 y2 42 4 x 56 2xy 2 3 a2b 57 Simplify each of the following expressing the final result as one radical For example 213 5 31 gt 2 1 gt 2 5 31 gt 4 5 2 4 3 a 2 3 12 b 2 3 1 4 3 c 2 3 1x3 d 21 3 x4 For Problems 17 32 perform the indicated operations and simplify Express final answers using positive ex ponents only Objective 2 17 3x 1 gt 4 5x1 gt 3 18 2x 2 gt 5 6x 1 gt 4 19 y 2 gt 3 y 21 gt 4 20 2x 1 gt 3 x 21 gt 2 21 4x 1 gt 4y 1 gt 2 3 22 5x 1 gt 2y 2 23 24x3y5 6x1y3 24 18x1y2 9x1y3 25 56a1y6 8a1y4 26 48b1y3 12b3y4 27 a2x1y3 3y1y4b 4 28 a6x2y5 7y2y3b 2 29 ax2 y3b 2 1y2 30 a a3 b2 2b 2 1y3 31 a 4a2x 2a1y2x1y3b 3 32 a 3ax2 1 a1y2x2 2b 2 For Problems 33 48 perform the indicated operations and express the answer in simplest radical form Objective 3 33 222 4 2 34 2 3 323 35 2 3 x2 4 x 36 2 3 x22 5 x3 37 2xy2 4 x3y5 38 2 3 x2y42 4 x3y 39 2 3 a2b22 4 a3b
85. 9 x 1 9 4x 1 1 x 1 3 4x 1 3 2x 1 1 2x 1 9 2x 1 3 2x 13 None of these possibilities yields a middle term of 6x Therefore 4x 2 1 6x 1 9 is not factorable using integers Certainly as the number of possibilities increases this trial and error technique for factoring becomes more tedious The key idea is to organize your work so that all pos sibilities are considered We have suggested one possible format in the previous ex amples However as you practice such problems you may devise a format that works better for you Whatever works best for you is the right approach There is another more systematic technique that you may wish to use with some trinomials It is an extension of the technique we used earlier with trinomials where the coefficient of the squared term was one To see the basis of this technique consider the following general product px 1 r qx 1 s 5 px qx 1 px s 1 r qx 1 r s 5 pq x 2 1 ps x 1 rq x 1 rs 5 pq x 2 1 ps 1 rq x 1 rs Notice that the product of the coefficient of x 2 and the constant term is pqrs Likewise the product of the two coefficients of x ps and rq is also pqrs Therefore the coeffi cient of x must be a sum of the form ps 1 rq such that the product of the coefficient of x 2 and the constant term is pqrs Now let s see how this works in some specific examples eXAMPLe 8 Factor 6x
86. All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 38 Together with our knowledge of dividing monomials these properties provide the basis for dividing polynomials by monomials Consider the following examples 18x 3 1 24x 2 6x 5 18x 3 6x 1 24x 2 6x 5 3x 2 1 4x 35x 2y 3 2 55x 3y4 5xy 2 5 35x 2y3 5xy 2 2 55x 3y4 5xy2 5 7xy 2 11x 2y2 Therefore to divide a polynomial by a monomial we divide each term of the polyno mial by the monomial As with many skills once you feel comfortable with the process you may then choose to perform some of the steps mentally Your work could take the following format 40x 4y 5 1 72x 5y7 8x 2y 5 5x 2y4 1 9x 3y 6 36a3b4 2 48a3b3 1 64a 2b 5 24a 2b 2 5 29ab 2 1 12ab 2 16b3 For Problems 1 8 answer true or false 1 The variables of a monomial term have exponents that are either positive integers or zero 2 The term 32xy2 is of degree 5 3 Any nonzero constant term is of degree zero 4 A polynomial is a monomial or a
87. B The following statements use the subset vocabulary and symbolism they are represented in Figure 0 1 Real numbers Whole numbers Natural numbers Integers Rational numbers Irrational numbers Figure 0 1 Real numbers Rational Irrational 2 1 Integers 2 0 1 Nonintegers 2 1 90360_ch0_001 102 indd 5 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 6 Unless otherwise noted all art on this page is Cengage Learning 1 The set of whole numbers is a subset of the set of integers 0 1 2 3 22 21 0 1 2 2 The set of integers is a subset of the set of rational numbers 22 21 0 1 2 x 0 x is a rational number 3 The set of rational numbers is a subset of the set of real numbers x 0 x is a rational number y 0 y is a real number Real Number Line and Absolute Value It is often helpful to have a geo
88. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 2 Exponents 25 Unless otherwise noted all art on this page is Cengage Learning eXAMPLe 8 Simplify 421 2 322 21 solution 421 2 322 21 5 a 1 41 2 1 32b 21 5 a1 4 2 1 9b 21 5 a 9 36 2 4 36b 21 5 a 5 36b 21 5 1 a 5 36b 1 5 36 5 Figure 0 15 shows calculator windows for Examples 7 and 8 Note that the answers are given in decimal form If your calculator also handles common fractions then the display window may appear as in Figure 0 16 Figure 0 15 Figure 0 16 eXAMPLe 9 Express a 21 1 b 22 as a single fraction involving positive exponents only solution a21 1 b22 5 1 a1 1 1 b2 5 a1 abab2 b2b 1 a 1 b2baa ab 5 b2 ab2 1 a ab2 5 b2 1 a ab2 classroom example Simplify 222 2 321 21 classroom example Express x23 1 y22 as a single fraction with positive exponent 90360_ch0_001 102 indd 25 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that an
89. Licensed to CengageBrain User Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing previous editions changes to current editions and alternate formats please visit www cengage com highered to search by ISBN author title or keyword for materials in your areas of interest Licensed to CengageBrain User 2013 2009 Brooks Cole Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced transmitted stored or used in any form or by any means graphic electronic or mechanical including but not limited to photocopying recording scanning digitizing taping Web distribution informat
90. age Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 2 Exponents 21 eXAMPLe 1 Find the indicated product 3x2y 4x3y2 solution 3x 2y 4x 3y 2 5 3 4 x 2 x 3 y y 2 5 12x 213y 112 bn bm 5 bn1m 5 12x 5y 3 eXAMPLe 2 Find the indicated product 22y3 5 solution 22y 3 5 5 22 5 y 3 5 ab n 5 an bn 5 232y 15 bn m 5 bmn eXAMPLe 3 Find the indicated quotient aa2 b4b 7 solution aa2 b4b 7 5 a2 7 b4 7 aa bb n 5 an bn 5 a14 b28 bn m 5 bmn eXAMPLe 4 Find the indicated quotient 256x9 7x4 solution 256x9 7x4 5 28x924 bn bm 5 bn2m when n m 5 28x5 Zero and Negative integers As exponents Now we can extend the concept of an exponent to include the use of zero and nega tive integers First let s consider the use of zero as an exponent We want to use zero in a way that Property 0 1 will continue to hold For example if bn bm 5 bn1m is to hold then classroom example Find the indicated pro
91. al expression 2 The rational expression 3x 2 4 x 1 2 is defined for all values of x 3 The rational expressions a2 2 4 b 2 2 and 24 2 a 2 2 2 b are equivalent 4 The quotient of any nonzero polynomial and its opposite is 21 5 To multiply rational expressions that do not have a common denominator we need to obtain equivalent fractions with a common denominator 6 Complex fractions are fractional forms that contain rational expressions in the numerator and or the denominator 7 The difference of 3x 2 4 7x 1 8 and 5x 2 1 7x 1 8 would equal zero if 3x 2 4 5 5x 2 1 8 Under what conditions would the product of x 1 2 x and x 2 2 x be equal to zero For Problems 1 18 simplify each rational expression Objective 1 1 14x2y 21xy 2 226xy2 65y 3 263xy4 281x2y 4 x2 2 y2 x2 1 xy 5 2x2y2 3 3xy 2 6 3a3b 2 6a2 b2 2 7 a2 1 7a 1 12 a2 2 6a 2 27 8 6x2 1 x 2 15 8x2 2 10x 2 3 9 2x3 1 3x2 2 14x x2y 1 7xy 2 18y 10 3x 2 x2 x2 2 9 11 x3 2 y3 x2 1 xy 2 2y2 12 ax 2 3x 1 2ay 2 6y 2ax 2 6x 1 ay 2 3y 13 2y 2 2xy x2y 2 y 14 16x3y 1 24x2y2 2 16xy3 24x2y 1 12xy2 2 12y3 15 8x2 1 4xy 2 2x 2 y 4x2 2 4xy 2 x 1 y 16 2x3 1 2y3 2x2 1 6x 1 2xy 1 6y 17 27x3 1 8y3 3x2 2 15x 1 2xy 2 10y 18 x3 1 64 3x2 1 11x 2 4 For Problems 19 68 perform the indicated operations involving rational expressions Express final answers in simplest form
92. al fourth roots of 16 are 2 and 22 2 Negative real numbers do not have real nth roots For example there are no real fourth roots of 216 90360_ch0_001 102 indd 65 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 66 If n is an odd positive integer greater than 1 then the following statements are true 1 Every real number has exactly one real nth root 2 The real nth root of a positive number is positive For example the fifth root of 32 is 2 3 The real nth root of a negative number is negative For example the fifth root of 232 is 22 In general the following definition is useful Definition 0 5 2 n b 5 a if and only if a n 5 b In Definition 0 5 if n is an even positive integer then a and b are both nonnegative If n is an odd positive integer greater than 1 then a and b are both nonnegative or both negative The symbol 1 n designates the principal root The following examples are applications of Definition
93. ational expressions 4 Simplify complex fractions Indicated quotients of algebraic expressions are called algebraic fractions or frac tional expressions The indicated quotient of two polynomials is called a rational expression This is analogous to de ning a rational number as the indicated quotient of two integers The following are examples of rational expressions 3x2 5 x 2 2 x 1 3 x2 1 5x 2 1 x2 2 9 xy2 1 x2y xy a3 2 3a2 2 5a 2 1 a4 1 a3 1 6 Because division by zero must be avoided no values can be assigned to variables that will create a denominator of zero Thus the rational expression x 2 2 x 1 3 is meaningful for all real number values of x except x 5 23 Rather than making restric tions for each individual expression we will merely assume that all denominators represent nonzero real numbers 90360_ch0_001 102 indd 51 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 52 The basic properties o
94. ause their product is 1 That is x 24 5 1 gt x 4 This suggests the following de nition Definition 0 4 If n is a positive integer and b is a nonzero real number then b2n 5 1 bn According to De nition 0 4 the following statements are true x25 5 1 x5 224 5 1 24 5 1 16 a3 4b 22 5 1 a3 4b 2 5 1 9 16 5 16 9 2 x23 5 2 1 x3 5 2x3 The rst four parts of Property 0 1 hold true for all integers Furthermore we do not need all three equations in part 5 of Property 0 1 The rst equation b n b m 5 b n2m 90360_ch0_001 102 indd 22 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 2 Exponents 23 can be used for all integral exponents Let s restate Property 0 1 as it pertains to inte gers We will include name tags for easy reference Property 0 2 If m and n are integers and a and b are real numbers with b 0 whenever it appears in a denominator then 1 bn bm 5 bn1m Product of two powers 2
95. b n m 5 b mn Power of a power 3 ab n 5 a nb n Power of a product 4 aa bb n 5 an bn Power of a quotient 5 b n b m 5 b n2m Quotient of two powers Having the use of all integers as exponents allows us to work with a large variety of numerical and algebraic expressions Let s consider some examples that illustrate the various parts of Property 0 2 eXAMPLe 5 Evaluate each of the following numerical expressions a 221 32 21 b a223 322b 22 solution a 221 32 21 5 221 21 32 21 Power of a product 5 21 322 Power of a power 5 2 a 1 32b 5 2a1 9b 5 2 9 b a223 322b 22 5 223 22 322 22 Power of a quotient 5 26 34 Power of a power 5 64 81 classroom example Evaluate each of the numerical expressions a 322 4 21 b a522 223b 22 90360_ch0_001 102 indd 23 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 24
96. bers 1 2 3 4 Natural numbers counting numbers positive integers 0 1 2 3 Whole numbers nonnegative integers 23 22 21 Negative integers 23 22 21 0 Nonpositive integers 22 21 0 1 2 Integers A rational number is de ned as any number that can be expressed in the form a b where a and b are integers and b is not zero The following are examples of rational numbers 2 3 23 4 21 7 9 2 6 1 2 because 6 1 2 5 13 2 24 because 24 5 24 1 5 4 21 0 because 0 5 0 1 5 0 2 5 0 3 etc 0 3 because 0 3 5 3 10 90360_ch0_001 102 indd 3 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 4 A rational number can also be de ned in terms of a decimal representation Before doing so let s brie y review the different possibilities for decimal representations Decimals can be classi ed as terminating repeating or non repeating Her
97. canned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 63 74 3 2 2 n 2 4 5 1 4 n 2 4 75 1 2 1 n 1 1 1 1 1 n 2 1 76 2 x 2 3 2 3 x 1 3 5 x2 2 9 2 2 x 2 3 77 22 x 2 4 x 1 2 3 x2 1 2x 1 3 x 78 21 y 2 2 1 5 x 3 x 2 4 xy 2 2x 79 1 1 x 1 1 1 x 80 2 2 x 3 2 2 x 81 a 1 a 1 4 1 1 82 3a 2 2 1 a 2 1 83 1 x 1 h 2 2 1 x2 h 84 1 x 1 h 3 2 1 x3 h 85 1 x 1 h 1 1 2 1 x 1 1 h 86 3 x 1 h 2 3 x h 87 2 2x 1 2h 2 1 2 2 2x 2 1 h 88 3 4x 1 4h 1 5 2 3 4x 1 5 h 89 x 21 1 2y 21 x 2 y 90 x 1 y x 21 1 y 21 91 x 1 2x21y22 4x21 2 3y22 92 x 22 2 2y 21 3x 21 1 y 22 69 Consider the addition problem 8 x 2 2 5 2 2 x Note that the denominators are opposites of each other If the property a 2 b 5 2 a b is applied to the second fraction we obtain 5 2 2 x 5 2 5 x 2 2 Thus we can proceed as follows 8 x
98. d or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 99 Apply rational exponents to simplify radical expressions section 0 7 Objective 3 To multiply or divide radical ex pressions with different indexes change from radical to exponen tial form Then apply the proper ties of exponents Finally change back to radical form Perform the indicated operation and express the answers in simplest radical form 2 4 xy32 5 x2y solution 2 4 xy32 5 x2y 5 x1y4y3y4 x2y5y1y5 5 x1y412y5y3y411y5 5 x13y20y19y20 5 2 20x13y19 Express the square root of a negative number in terms of i section 0 8 Objective 1 We can represent a square root of any negative real number as the product of a real number and the imaginary unit i That is 22b 5 i2b where b is a pos itive real number Write 2248 in terms of i and simplify solution 2248 5 221248 5 i21623 5 4i23 OBjecTiVe sUMMARy eXAMPLe Add and subtract complex numbers section 0 8 Objective 2 We describe the addition and subtraction of complex numbers as follows a 1 bi 1 c 1 di 5 a 1 c 1 b 1 d
99. d 4 2 x2 are opposites The next example illustrates how we use this idea when simplifying rational expressions 4 2 x2 x2 1 x 2 6 5 2 1 x x 1 3 2 2 x x 2 2 5 21 ax 1 2 x 1 3b 2 2 x x 2 2 5 21 5 2x 1 2 x 1 3 or 2x 2 2 x 1 3 Multiplying and Dividing Rational expressions Multiplication of rational expressions is based on the following property a b c d 5 ac bd In other words we multiply numerators and we multiply denominators and express the final product in simplified form Study the following examples carefully and pay spe cial attention to the formats used to organize the computational work y 3x 4y 8y2 9x 5 3 8 2 x y2 4 9 3 x y 5 2y 3 x 2 12x2y 218xy 224xy2 56y3 5 12 2 24 3 x3 y3 18 3 56 7 x y4 5 2x2 7y 12x2y 218xy 5 2 12x2y 18xy and 224xy2 56y3 5 2 24xy2 56y3 y so the product is positive y x2 2 4 x 1 2 y2 5 y x 1 2 y2 x 1 2 x 2 2 5 1 y x 2 2 y x2 2 x x 1 5 x2 1 5x 1 4 x4 2 x2 5 x x 2 1 x 1 1 x 1 4 x 1 5 x2 x 1 1 x 2 1 5 x 1 4 x x 1 5 x 90360_ch0_001 102 indd 53 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial
100. dex and the same radicand 322 1 522 5 3 1 5 22 5 822 72 3 5 2 32 3 5 5 7 2 3 2 3 5 5 42 3 5 Sometimes it is necessary to simplify the radicals first and then to combine them by applying the distributive property 328 1 2218 2 422 5 32422 1 22922 2 422 5 622 1 622 2 422 5 6 1 6 2 4 22 5 822 Multiplying Radicals Property 0 3 can also be viewed as 2 n b2 n c 5 2 n bc Then along with the commutative and associative properties of the real numbers it provides the basis for multiplying radicals that have the same index Consider the following two examples A726BA328B 5 7 3 26 28 5 21248 5 2121623 90360_ch0_001 102 indd 67 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 68 5 21 4 23 5 8423 A22 3 6BA52 3 4B 5 2 5 2 3 6 2 3 4 5 102 3 24 5 102 3 82 3 3 5 10 2 2 3 3 5 202 3 3 The distributive property along with Property 0 3 provides a way of
101. duct 5a3b2 22ab4 classroom example Find the indicated product 23x2 4 classroom example Find the indicated quotient ax5 y b 3 classroom example Find the indicated quotient 21a8 23a3 90360_ch0_001 102 indd 21 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 22 x4 x0 should equal x 410 which equals x 4 In other words x 0 acts like 1 because x4 x0 5 x4 Look at the following de nition Definition 0 3 If b is a nonzero real number then b0 5 1 Therefore according to De nition 0 3 the following statements are all true 50 5 1 2413 0 5 1 a 3 11b 0 5 1 x3y4 0 5 1 if x 0 and y 0 A similar line of reasoning can be used to motivate a de nition for the use of negative integers as exponents Consider the example x4 x24 If bn bm 5 bn1m is to hold then x4 x24 should equal x 41 24 which equals x0 5 1 Therefore x 24 must be the reciprocal of x 4 bec
102. e 8 The associative properties are grouping properties 9 On the rectangular coordinate system the point of intersection of the two axes is called the origin 10 The horizontal axis is customarily referred to as the y axis Remark You can find answers to the Concept Quiz questions at the end of the next Problem Set concept Quiz 0 1 90360_ch0_001 102 indd 16 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 17 For Problems 11 18 list those elements of the set of num bers e 0 25 222 7 8 2 10 13 71 8 0 279 0 467 2p 214 46 6 75 f that belong to each of the following sets Objective 2 11 The natural numbers 12 The whole numbers 13 The integers 14 The rational numbers 15 The irrational numbers 16 The nonnegative integers 17 The nonpositive integers 18 The real numbers For Problems 19 32 use the following set designations N 5 5x0x is a natural number6 W 5 5x0x is a whole number6 I 5 5x0x is an i
103. e the commutative and asso ciative properties of addition hold for all complex numbers The additive identity ele ment is 0 1 0i or simply the real number 0 The additive inverse of a 1 bi is 2a 2 bi because a 1 bi 1 2a 2 bi 5 a 1 2a 1 b 1 2b i 5 0 Therefore to subtract c 1 di from a 1 bi we add the additive inverse of c 1 di a 1 bi 2 c 1 di 5 a 1 bi 1 2c 2 di 5 a 2 c 1 b 2 d i 90360_ch0_001 102 indd 83 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 84 The following examples illustrate the subtraction of complex numbers 9 1 8i 2 5 1 3i 5 9 2 5 1 8 2 3 i 5 4 1 5i 3 2 2i 2 4 2 10i 5 3 2 4 1 22 2 210 i 5 21 1 8i a2 1 2 1 1 3 ib 2 a3 4 1 1 2 ib 5 a2 1 2 2 3 4b 1 a1 3 2 1 2b i 5 2 5
104. e are doing at this time Exponents can also be used to indicate repeated multiplication of polynomi als For example 3x 2 4y 2 means 3x 2 4y 3x 2 4y and x 1 4 3 means x 1 4 x 1 4 x 1 4 Therefore raising a polynomial to a power is merely another multiplication problem 3x 2 4y 2 5 3x 2 4y 3x 2 4y 5 9x 2 2 24xy 1 16y 2 90360_ch0_001 102 indd 34 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 3 Polynomials 35 Hint When squaring a binomial be careful not to forget the middle term That is x 1 5 2 x 2 1 25 instead x 1 5 2 5 x 2 1 10x 1 25 x 1 4 3 5 x 1 4 x 1 4 x 1 4 5 x 1 4 x 2 1 8x 1 16 5 x x 2 1 8x 1 16 1 4 x 2 1 8x 1 16 5 x 3 1 8x 2 1 16x 1 4x 2 1 32x 1 64 5 x 3 1 12x 2 1 48x 1 64 special Patterns In multiplying binomials you should learn to recognize some special patterns These patterns can be used to nd products and some of them will be helpful later when you are factoring polynomials a 1 b
105. e are some examples of each 0 3 0 46 0 789 0 2143 Terminating decimals 0 333 p 0 1414 p 0 7127127 p 0 241717 p Repeating decimals 0 472195631 p 0 21411711191111 p 3 141592654 p 1 414213562 p Nonrepeating decimals A repeating decimal has a block of digits that repeats inde nitely This repeating block of digits may be of any size and may or may not begin immediately after the decimal point A small horizontal bar is commonly used to indicate the repeating block Thus 0 3333 can be expressed as 0 3w and 0 24171717 as 0 2417 w In terms of decimals a rational number is de ned as a number with either a termi nating or a repeating decimal representation The following examples illustrate some rational numbers written in a b form and in the equivalent decimal form 3 4 5 0 75 3 11 5 0 27 1 8 5 0 125 1 7 5 0 142857 1 3 5 0 3 We de ne an irrational number as a number that cannot be expressed in a b form form where a and b are integers and b is not zero Furthermore an irrational number has a nonrepeating nonterminating decimal representation Following are some examples of irrational numbers and a partial decimal representation for each number Note that the decimals do not terminate and do not repeat 12 5 1 414213562373095 p 13 5 1 73205080756887 p p 5 3 14159265358979 p The entire set of real numbers is composed of the rational numbers along wi
106. e overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 8 Complex Numbers 87 4 2 5i 2i 5 4 2 5i 2i 22i 22i 5 4 2 5i 22i 2i 22i 5 28i 1 10i2 24i2 5 28i 1 10 21 24 21 5 210 2 8i 4 5 2 5 2 2 2i For a problem such as the last one in which the denominator is a pure imaginary num ber we can change to standard form by choosing a multiplier other than the conjugate of the denominator Consider the following alternative approach 4 2 5i 2i 5 4 2 5i 2i i i 5 4 2 5i i 2i i 5 4i 2 5i2 2i2 5 4i 2 5 21 2 21 5 5 1 4i 22 5 2 5 2 2 2i For Problems 1 8 answer true or false 1 The number i is not a real number 2 The number i2 is a real number 3 The form ai 1 b is called the standard form of a complex number 4 Every real number is a member of the set of complex numbers 5 The principal square root of any negative real number can be represented as the product of a real number and the imaginary unit i 6 6 2 4i and 26 1 4i are additive inverses 7 The conjugate of the number 22 2 3i is 2 1 3i 8 The product of a complex number and its conjugate is a real number concept Quiz 0 8 90360_ch0_001 102 indd 87 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied
107. e the various forms used to indicate Operation Arithmetic Algebra Vocabulary Addition 4 1 6 x 1 y The sum of x and y Subtraction 14 2 10 a 2 b The difference of a and b Multiplication 7 3 5 or 7 5 a b a b a b a b or ab The product of a and b Division 8 4 4 8 4 8 gt 4 or 4q8 x 4 y x y x gt y or yqx y 0 The quotient of x divided by y 0 1 Some Basic Ideas O B j e c T i V e s 1 Recognize the vocabulary and symbolism associated with sets 2 Know the various subset classifications of the real number system 3 Find distance on a number line 4 Apply the definition of the absolute value of a number 5 Know the real number properties 6 Evaluate algebraic expressions 7 Review the Cartesian coordinate system 90360_ch0_001 102 indd 2 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 3 division In algebra the fraction forms x y and x gt y are generally used although the other forms do serve a purpose at times
108. e two complex numbers a 1 bi and a 2 bi are called conjugates of each other and the product of a complex number and its conjugate is a real number This can be shown as follows a 1 bi a 2 bi 5 a a 2 bi 1 bi a 2 bi 5 a2 2 abi 1 abi 2 b2i2 5 a2 2 b2 21 5 a2 1 b2 Conjugates are used to simplify an expression such as 3i gt 5 1 2i which indi cates the quotient of two complex numbers To eliminate i in the denominator and to change the indicated quotient to the standard form of a complex number we can multiply both the numerator and denominator by the conjugate of the denominator 3i 5 1 2i 5 3i 5 1 2i 5 2 2i 5 2 2i 5 3i 5 2 2i 5 1 2i 5 2 2i 5 15i 2 6i2 25 2 4i2 5 15i 2 6 21 25 2 4 21 5 6 1 15i 29 5 6 29 1 15 29 i The following examples further illustrate the process of dividing complex numbers 2 2 3i 4 2 7i 5 2 2 3i 4 2 7i 4 1 7i 4 1 7i 5 2 2 3i 4 1 7i 4 2 7i 4 1 7i 5 8 1 14i 2 12i 2 21i2 16 2 49i2 5 8 1 2i 2 21 21 16 2 49 21 5 29 1 2i 65 5 29 65 1 2 65 i 90360_ch0_001 102 indd 86 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect th
109. eXAMPLe 6 Find the indicated products and quotients and express the nal results with positive integral exponents only a 3x 2y 24 4x 23y b 12a3b2 23a21b5 c a15x21y2 5xy24 b 21 solution a 3x2y24 4x23y 5 12x21 23 y2411 Product of powers 5 12x21y23 5 12 xy3 b 12a3b2 23a21b5 5 24a32 21 b225 Quotient of powers 5 24a4b23 5 24a4 b3 c a15x21y2 5xy24 b 21 5 3x2121y22 24 21 First simplify inside parentheses 5 3x22y6 21 5 321x2y26 Power of a product 5 x2 3y6 The next two examples illustrate the simpli cation of numerical and algebraic expressions involving sums and differences In such cases De nition 0 4 can be used to change from negative to positive exponents so that we can proceed in the usual ways eXAMPLe 7 Simplify 223 1 321 solution 223 1 321 5 1 23 1 1 31 5 1 8 1 1 3 5 3 24 1 8 24 5 11 24 classroom example Find the indicated products and quotients and express the final results with positive integral exponents only a 5a23b21c23 3a22bc7 b 228x2y3 7x4y22 c a12a3b22 3a4b23 b 21 classroom example Simplify 421 1 221 90360_ch0_001 102 indd 24 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s
110. ecessary or per haps the same technique used twice Factor 81a4 2 16b4 solution 81a4 2 16b4 5 9a2 1 4b2 9a2 2 4b2 5 9a2 1 4b2 3a 1 2b 3a 2 2b OBjecTiVe sUMMARy eXAMPLe Simplify rational expres sions section 0 5 Objective 1 A rational expression is defined as the indicated quotient of two polynomials The Fundamental Principle of Fractions a k b k 5 a b is used when reducing rational numbers or rational expressions Simplify x2 2 2x 2 15 x2 1 x 2 6 solution x2 2 2x 2 15 x2 1 x 2 6 5 x 1 3 x 2 5 x 1 3 x 2 2 5 x 2 5 x 2 2 continued Chapter 0 Summary 90360_ch0_001 102 indd 95 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 96 Multiply and divide ra tional expressions section 0 5 Objective 2 Multiplication of rational expres sions is based on the following definition a b c d 5 ac bd Division of rational expressions is based
111. ed as follows 3 x 1 2 y 5 x 2 6 y2 5 a3 xbay yb 1 a2 ybax xb a5 xbay2 y2b 2 a 6 y2bax xb 5 3y xy 1 2x xy 5y2 xy2 2 6x xy2 5 3y 1 2x xy 5y2 2 6x xy2 y 5 3y 1 2x xy xy2 5y2 2 6x 5 y 3y 1 2x 5y2 2 6x classroom example Simplify 4 a 2 5 b 3 a 1 6 b2 90360_ch0_001 102 indd 58 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 59 solution B The LCD of all four denominators x y x and y 2 is xy 2 Let s multiply the entire complex fraction by a form of 1 namely xy 2 gt xy 2 3 x 1 2 y 5 x 2 6 y2 5 3 x 1 2 y 5 x 2 6 y2 axy2 xy2b 5 xy2 a3 xb 1 xy2 a2 yb xy2 a5 xb 2 xy2 a 6 y2b 5 3y2 1 2xy 5y2 2 6x or y 3y 1 2x 5y2 2 6x Certainly either approach Solution A or Solution B will work with a problem such as Example 7 We suggest that you study Solution B very carefully
112. ed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 48 Unless otherwise noted all art on this page is Cengage Learning equals the given polynomial We can also give some visual support to a factoring prob lem by graphing the given polynomial and its completely factored form on the same set of axes as shown for Example 10 in Figure 0 19 Note that the graphs for Y1 5 24x2 1 2x 2 15 and Y2 5 6x 1 5 4x 2 3 appear to be identical 20 220 5 25 Figure 0 19 sum and Difference of Two cubes Earlier in this section we discussed the difference of squares factoring pattern We pointed out that no analogous sum of squares pattern exists that is a polynomial such as x2 1 9 is not factorable using integers However there do exist patterns for both the sum and the difference of two cubes These patterns come from the following special products x 1 y x 2 2 xy 1 y 2 5 x x 2 2 xy 1 y 2 1 y x 2 2 xy 1 y2 5 x 3 2 x 2y 1 xy 2 1 x 2y 2 xy 2 1 y 3 5 x 3 1 y 3 x 2 y x 2 1 xy 1 y 2 5 x x 2 1 xy 1 y 2 2 y x 2 1 xy 1 y 2 5 x 3 1 x 2y 1 xy 2 2 x 2y 2 xy 2 2 y 3 5
113. ent does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 2 Let s begin by pulling together the basic tools we need for the study of algebra In arithmetic symbols such as 6 2 3 0 27 and p are used to represent numbers The operations of addition subtraction multiplication and division are commonly indi cated by the symbols 1 2 3 and 4 respectively These symbols enable us to form speci c numerical expressions For example the indicated sum of 6 and 8 can be writ ten 6 1 8 In algebra we use variables to generalize arithmetic ideas For example by using x and y to represent any two numbers we can use the expression x 1 y to represent the indicated sum of any two numbers The x and y in such an expression are called vari ables and the phrase x 1 y is called an algebraic expression Many of the notational agreements we make in arithmetic can be extended to alge bra with a few modi cations The following chart summarizes those notational agree ments regarding the four basic operations Note the different ways of indicating a product including the use of parentheses The ab form is the simplest and probably the most widely used form Expressions such as abc 6xy and 14xyz all indicate multiplication Notic
114. ent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 82 So far we have dealt only with real numbers However as we get ready to solve equa tions in the next chapter there is a need for more numbers There are some very simple equations that do not have solutions if we restrict ourselves to the set of real numbers For example the equation x 2 1 1 5 0 has no solutions among the real numbers To solve such equations we need to extend the real number system In this section we will introduce a set of numbers that contains some numbers with squares that are negative real numbers Then in the next chapter and in Chapter 4 we will see that this set of numbers called the complex numbers provides solutions not only for equations such as x 2 1 1 5 0 but also for any polynomial equation in general Let s begin by defining a number i such that i 2 5 21 The number i is not a real number and is often called the imaginary unit but the num ber i 2 is the real number 21 The imaginary unit i is used to define a complex number as follows Definition 0 8 A complex number is any number that can be expressed in the form a 1 bi where a and b are real numbers and i is the imaginary unit The form a 1 bi is called the standard form of a complex number The real num ber a is called the real part of the complex number and b is called the imaginary part
115. er to these number lines as the horizontal axis and the vertical axis or together as the coordinate axes They partition a plane into four regions called quadrants The quadrants are numbered counterclockwise from I through IV as indicated in Figure 0 8 The point of intersection of the two axes is called the origin classroom example Evaluate 2 22y 2 1 2 2 y 1 4 when y 5 22 90360_ch0_001 102 indd 12 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 13 Unless otherwise noted all art on this page is Cengage Learning I II IV III Figure 0 8 The positive direction on the horizontal axis is to the right and the positive direc tion on the vertical axis is up It is now possible to set up a one to one correspondence between ordered pairs of real numbers and the points in a plane To each ordered pair of real numbers there corresponds a unique point in the plane and to each point in the plane there corresponds a unique ordered pair of real numbers A par
116. eview has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 92 Evaluate algebraic expressions section 0 1 Objective 6 An algebraic expression takes on a numerical value whenever each variable in the expression is replaced by a real number It is good practice to use paren theses when replacing the varia ble with a number Evaluate a 2 4b a 2 b 2 when a 5 4 and b 5 21 solution a 2 4b a 2 b 2 5 4 2 4 2 1 4 2 2 1 2 5 4 1 4 5 2 5 8 25 when a 5 4 and b 5 21 Apply the properties of ex ponents to simplify alge braic expressions section 0 2 Objective 2 Read Property 0 2 on page 23 A quick summary of some of that information is as follows 1 When multiplying like bases add the exponents 2 When dividing like bases subtract the exponents 3 When a power is raised to an other power multiply the ex ponents Simplify a 6x3y24 3x22y21b 2 solution a 6x3y24 3x22y21b 2 5 2x32 22 y242 21 2 5 2x5y23 2 5 22x10y26 5 4x10 y6 Write numbers in scientific notation section 0 2 Objective 3 Scientific notation is often used to write numbers that are very small or very large in magni tude The scient
117. examples illustrate De nition 0 2 23 5 2 2 2 5 8 a1 2b 5 5 1 2 1 2 1 2 1 2 1 2 5 1 32 34 5 3 3 3 3 5 81 0 7 2 5 0 7 0 7 5 0 49 25 2 5 25 25 5 25 252 5 2 5 5 5 225 97 Different graphing calculators use different se quences of key strokes to evaluate algebraic ex pressions Be sure that you can do Problems 59 80 with your calculator Graphing calculator Activities Answers to the concept Quiz 1 False 2 True 3 True 4 False 5 False 6 False 7 False 8 True 9 True 10 False 0 2 Exponents O B j e c T i V e s 1 Evaluate numerical expressions that have integer exponents 2 Apply the properties of exponents to simplify algebraic expressions 3 Write numbers in scientific notation 4 Convert numbers from scientific notation to ordinary decimal notation 5 Perform calculations with numbers in scientific form 90360_ch0_001 102 indd 19 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0
118. f the real numbers can be used for working with rational expressions For example the property a k b k 5 a b which is used to reduce rational numbers is also used to simplify rational expressions Consider the following examples 15xy 25y 5 3 5 x y 5 5 y 5 3x 5 29 18x2y 5 2 9 1 18 2 x2y 5 2 1 2x2y Note that slightly different formats were used in these two examples In the first one we factored the coefficients into primes and then proceeded to simplify however in the second problem we simply divided a common factor of 9 out of both the numerator and denominator This is basically a format issue and depends on your personal preference Also notice that in the second example we applied the property 2a b 5 2a b This is part of the general property that states 2a b 5 a 2b 5 2a b The properties bn m 5 bmn and ab n 5 anbn may also play a role when simplifying a rational expression as the next example demonstrates 4x3y 2 6x y2 2 5 42 x3 2 y2 6 x y4 5 16 8 x6y2 6 3xy4 5 8x5 3y2 The factoring techniques discussed in the previous section can be used to factor numerators and denominators so that the property a k gt b k 5 a gt b can be applied Consider the following examples x2 1 4x x2 2 16 5 x x 1 4 x 2 4 x 1 4 5 x x 2 4 5n2 1 6n 2 8 10n2 2 3n 2 4 5 5n 2 4 n 1 2 5n 2 4 2n 1 1 5 n 1 2
119. ght 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 5 Rational Expressions 57 Unless otherwise noted all art on this page is Cengage Learning 8 x x 2 4 1 2 x 5 8 x x 2 4 1 a2 xbax 2 4 x 2 4b 5 8 x x 2 4 1 2 x 2 4 x x 2 4 5 8 1 2x 2 8 x x 2 4 5 2x x x 2 4 5 2 x 2 4 In Figure 0 20 we give some visual support for our answer in Example 5 by graph ing Y1 5 8 x2 2 4x 1 2 x and Y2 5 2 x 2 4 Certainly their graphs appear to be identical but a word of caution is needed here Actually the graph of Y1 5 8 x2 2 4x 1 2 x has a hole at a0 2 1 2b because x cannot equal zero When you use a graphing calculator this hole may not be detected Except for the hole the graphs are identical and we are claiming that 8 x2 2 4x 1 2 x 5 2 x 2 4 for all values of x except 0 and 4 10 210 10 210 Figure 0 20 eXAMPLe 6 Add 3n n2 1 6n 1 5 1 4 n2 2 7n 2 8 solution n2 1 6n 1 5 5 n 1 5 n 1 1 n2 2 7n 2
120. gnitude of the problem Let s use both forms on the fol lowing two problems 82y3 5 2 3 82 5 2 3 64 5 4 or 82y3 5 A 2 3 8B2 5 2 2 5 4 272y3 5 2 3 272 5 2 3 729 5 9 or 272y3 5 A 2 3 27B2 5 3 2 5 9 90360_ch0_001 102 indd 76 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 7 Relationship Between Exponents and Roots 77 To compute 82 gt 3 both forms work equally well However to compute 272 gt 3 the form A 2 3 27B2 is much easier to handle The following examples further illustrate Definition 0 7 253y2 5 A 225B3 5 53 5 125 32 22y5 5 1 32 2y5 5 1 A 2 5 32B2 5 1 22 5 1 4 264 2y3 5 A 2 3 264B2 5 24 2 5 16 284y3 5 2A 2 3 8B4 5 2 2 4 5 216 It can be shown that all of the results pertaining to integral exponents listed in Property 0 2 on page 23 also hold for all rational exponents Let s consider some examples to illustrate each of those results x1y2 x2y3 5 x1y212y3 bn bm 5 bn1m 5 x3y614y6 5 x7y6 a2y3 3y2 5 a 3y2 2y3 bn
121. h illustrate the process of nding a value of an algebraic expression The process is commonly referred to as evaluating an alge braic expression eXAMPLe 1 Find the value of 3xy 2 4z when x 5 2 y 5 24 and z 5 25 solution 3xy 2 4z 5 3 2 24 2 4 25 when x 5 2 y 5 24 and z 5 25 3xy 2 4z 5 224 1 20 3xy 2 4z 5 24 eXAMPLe 2 Find the value of a 2 4b 2 2c 1 1 when a 5 28 b 5 27 and c 5 14 solution a 2 4b 2 2c 1 1 5 28 2 4 27 2 2 14 1 1 a 2 4b 2 2c 1 1 5 28 2 228 2 29 a 2 4b 2 2c 1 1 5 28 2 257 a 2 4b 2 2c 1 1 5 49 eXAMPLe 3 Evaluate a 2 2b 3c 1 5d when a 5 14 b 5 212 c 5 23 and d 5 22 solution a 2 2b 3c 1 5d 5 14 2 2 212 3 23 1 5 22 5 14 1 24 29 2 10 5 38 219 5 22 classroom example Find the value of 22a 1 4bc when a 5 23 b 5 5 and c 5 21 classroom example Find the value of 3 2x 2 5y 2 4 when x 5 21 and y 5 26 classroom example Evaluate 2x 2 3y x 2 y when x 5 24 and y 5 2 90360_ch0_001 102 indd 11 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequen
122. h neither the numerator nor the denominator of the radicand is a perfect nth power B 2 3 5 22 23 5 22 23 23 23 5 26 3 Form of 1 The process used to simplify the radical in this example is referred to as rationalizing the denominator There is more than one way to rationalize the denominator as il lustrated by the next example 90360_ch0_001 102 indd 69 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 70 eXAMPLe 1 Simplify 25 28 solution A 25 28 5 25 28 28 28 5 240 8 5 24210 8 5 2210 8 5 210 4 solution B 25 28 5 25 28 22 22 5 210 216 5 210 4 solution c 25 28 5 25 2422 5 25 222 5 25 222 22 22 5 210 4 The three approaches in Example 1 again illustrate the need to think first and then push the pencil You may find one approach easier than another eXAMPLe 2 Simplify 26
123. her Bill Smith Group Copy Editor Graphic World Inc Illustrator Network Graphics Graphic World Inc Cover Designer Irene Morris Cover Image Fotolia Compositor Graphic World Inc For product information and technology assistance contact us at Cengage Learning Customer amp Sales Support 1 800 354 9706 For permission to use material from this text or product submit all requests online at www cengage com permissions Further permissions questions can be e mailed to permissionrequest cengage com Printed in the United States of America 1 2 3 4 5 6 7 16 15 14 13 12 90360_fm_i xiv indd 4 11 17 11 9 02 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User The temperature in Big Lake Alaska at 3 p m was 24 F By 11 p m the temperature had dropped another 20 We can use the numerical expression 24 2 20 to determine the temperature at 11 p m Megan has p pennies n nickels d dimes and q quarters The algebraic expression p 1 5n 1 10d 1 25q can be used to represent the total amount
124. hird party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 1 Evaluate each of the following a 2722 b a3 2b 23 c a4 9b 3y2 d B 3 27 64 2 Find the product 23x21y2 5x23y24 and express the result using positive exponents only For Problems 3 7 perform the indicated operations 3 23x 2 4 2 7x 2 5 1 22x 2 9 4 5x 2 2 26x 1 4 5 x 1 2 3x 2 2 2x 2 7 6 4x 2 1 3 7 218x4y3 2 24x5y4 22xy2 For Problems 8 11 factor each polynomial completely 8 18x 3 2 15x 2 2 12x 9 30x 2 2 13x 2 10 10 8x 3 1 64 11 x 2 1 xy 2 2y 2 2x For Problems 12 16 perform the indicated operations involving rational expressions Express nal answers in simplest form 12 6x3y2 5xy 4 8y 7x3 13 x2 2 4 2x2 1 5x 1 2 2x2 1 7x 1 3 x3 2 8 14 3n 2 2 4 2 4n 1 1 6 15 5 2x2 2 6x 1 4 3x2 1 6x 16 4 n2 2 3 2n 2 5 n 17 Simplify the complex fraction 2 x 2 5 y 3 x 1 4 y2 For Problems 18 21 express each radical expression in simplest radical form All variables represent positive real numbers 18 6228x5 19 526 3212 20 26 222
125. ht 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 42 parentheses can be factored further Moreover in the last form 1 2 6x 2 1 24x the condition of using only integers is violated This application of the distributive property is often referred to as factoring out the highest common monomial factor The following examples illustrate the process 12x 3 1 16x 2 5 4x 2 3x 1 4 8ab 2 18b 5 2b 4a 2 9 6x 2y 3 1 27xy 4 5 3xy 3 2x 1 9y 30x 3 1 42x 4 2 24x 5 5 6x 3 5 1 7x 2 4x 2 Sometimes there may be a common binomial factor rather than a com mon monomial factor For example each of the two terms in the expression x y 1 2 1 z y 1 2 has a binomial factor of y 1 2 Thus we can factor y 1 2 from each term and obtain the following result x y 1 2 1 z y 1 2 5 y 1 2 x 1z Consider a few more examples involving a common binomial factor a 2 b 1 1 1
126. ide of zero 3 Locate the opposite of 2x written as 2 2x on the other side of zero Figure 0 5 b a c d Figure 0 4 90360_ch0_001 102 indd 6 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 7 Unless otherwise noted all art on this page is Cengage Learning Therefore we conclude that the opposite of the opposite of any real number is the number itself and we express this symbolically by 2 2x 5 x Remark The symbol 21 can be read negative one the negative of one the oppo site of one or the additive inverse of one The opposite of and additive inverse of terminology is especially meaningful when working with variables For example the symbol 2x read the opposite of x or the additive inverse of x emphasizes an important issue Because x can be any real number 2x opposite of x can be zero positive or negative If x is positive then 2x is negative If x is negative then 2x is positive If x is zero then
127. ific form of a number is expressed as N 10k where N is a number greater than or equal to 1 and less than 10 written in decimal form and k is an integer Write each of the following in scientific notation a 0 00000342 b 678 000 000 000 solution a 0 00000342 5 3 42 1026 b 678 000 000 000 5 6 78 1011 OBjecTiVe sUMMARy eXAMPLe Convert numbers from scientific notation to ordinary decimal notation section 0 2 Objective 4 To switch from scientific nota tion to ordinary decimal nota tion move the decimal point the number of places indicated by the exponent of the 10 The dec imal point is moved to the right if the exponent is positive and to the left if the exponent is negative Write each of the following in ordinary deci mal notation a 8 5 1025 b 3 4 106 solution a 8 5 1025 5 0 000085 b 3 4 106 5 3 400 000 90360_ch0_001 102 indd 92 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 93 Perform calculations with numbers in
128. in User 0 2 Exponents 29 59 x25 x22 60 a23 a5 61 a2b23 a21b22 62 x21y22 x3y21 For Problems 63 70 nd the indicated products quo tients and powers express answers without using zero or negative integers as exponents Objective 2 63 4x 3y 2 25xy 3 64 26xy 3x 2y 4 65 23xy 3 3 66 22x 2y 4 4 67 a2x2 3y3b 3 68 a 4x 5y2b 3 69 72x8 29x2 70 108x6 212x2 For Problems 71 80 nd the indicated products and quotients express results using positive integral expo nents only Objective 2 71 2x 21y 2 3x 22y 23 72 4x 22y 3 25x 3y 24 73 26a 5y 24 2a 27y 74 28a 24b 25 26a 21b 8 75 24x21y22 6x24y3 76 56xy23 8x2y2 77 235a3b22 7a5b21 78 27a24b25 23a22b24 79 a14x22y24 7x23y26 b 22 80 a 24x5y23 28x6y21b 23 For Problems 81 88 express each as a single fraction involving positive exponents only Objective 2 81 x21 1 x22 82 x22 1 x24 83 x 22 2 y 21 84 2x 21 2 3y 23 85 3a 22 1 2b 23 86 a 22 1 a 21b 22 87 x 21y 2 xy 21 88 x 2y 21 2 x 23y 2 23 42 21 24 321 3 25 321 22 21 26 23 322 22 27 42 521 2 28 222 421 3 29 a222 521b 22 30 a321 223b 22 31 a322 821b 2 32 a 42 521b 21 33 23 223 34 223 23 35 1021 104 36 1023 1027 37 322 1 223 38 223 1 521 39 a2 3b 21 2 a3 4b 21
129. in simplest form because the radicand contains a perfect power of the index Thus we simplified 2 3 x4 by expressing it as 2 3 x31 3 x which in turn can be written x1 3 x Such simplification can also be done in exponential form as follows 2 3 x4 5 x 4 gt 3 5 x 3 gt 3 x 1 gt 3 5 x x 1 gt 3 5 x1 3 x Note the use of this type of simplification in the following examples eXAMPLe 1 Perform the indicated operations and express the answers in simplest radical form a 2 3 x22 4 x3 b 222 3 4 c 227 2 3 3 solutions a 2 3 x22 4 x3 5 x2y3 x3y4 5 x2y313y4 5 x17y12 5 x12y12 x5y12 5 x2 12 x5 b 222 3 4 5 21 gt 2 41 gt 3 5 21 gt 2 22 1 gt 3 5 21 gt 2 22 gt 3 5 21 gt 212 gt 3 5 27 gt 6 5 26 gt 6 21 gt 6 5 22 6 2 c 227 2 3 3 5 271y2 31y3 5 33 1y2 31y3 5 33y2 31y3 5 33y221y3 5 37y6 5 36 gt 6 31 gt 6 5 32 6 3 The process of rationalizing the denominator can sometimes be handled more easily in exponential form Consider the following examples which illustrate this procedure eXAMPLe 2 Rationalize the denominator and express the answer in simplest radical form a 2 1 3 x b 1 3 x 1y solutions a 2 2 3 x 5 2 x1y3 5 2 x1y3 x2y3 x2y3 5 2x2y3 x 5 22 3 x2 x b 2 3 x 2y 5 x1y3 y1y2 5 x1y3 y1y2 y1y2 y1y2 5 x1y3 y1y2 y 5 x2y6 y3y6 y 5 2 6 x
130. ion networks or information storage and retrieval systems except as permitted under Section 107 or 108 of the 1976 United States Copyright Act without the prior written permission of the publisher Library of Congress Control Number 2011938660 ISBN 13 978 1 111 99036 7 ISBN 10 1 111 99036 0 Brooks Cole 20 Davis Drive Belmont CA 94002 3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe including Singapore the United Kingdom Australia Mexico Brazil and Japan Locate your local office at www cengage com global Cengage Learning products are represented in Canada by Nelson Education Ltd To learn more about Brooks Cole visit www cengage com brookscole Purchase any of our products at your local college store or at our preferred online store www CengageBrain com College Algebra Eighth Edition Jerome E Kaufmann and Karen L Schwitters Acquisitions Editor Gary Whalen Developmental Editor Assistant Editor Cynthia Ashton Editorial Assistant Sabrina Black Media Editor Lynh Pham Senior Marketing Manager Danae April Marketing Communications Manager Mary Anne Payumo Content Project Manager Jennifer Risden Design Director Rob Hugel Art Director Vernon Boes Manufacturing Planner Becky Cross Rights Acquisitions Specialist Roberta Broyer Production Service Graphic World Inc Text Designer Diane Beasley Photo Researc
131. ith 21 we can simplify and express the final product in the standard form of a complex number Consider the following examples 2 1 3i 4 1 5i 5 2 4 1 5i 1 3i 4 1 5i 5 8 1 10i 1 12i 1 15i2 5 8 1 22i 1 15 21 5 8 1 22i 2 15 5 27 1 22i 1 2 7i 2 5 1 2 7i 1 2 7i 5 1 1 2 7i 2 7i 1 2 7i 5 1 2 7i 2 7i 1 49i2 5 1 2 14i 1 49 21 5 1 2 14i 2 49 5 248 2 14i 2 1 3i 2 2 3i 5 2 2 2 3i 1 3i 2 2 3i 5 4 2 6i 1 6i 2 9i2 5 4 2 9 21 5 4 1 9 5 13 Remark Don t forget that when multiplying complex numbers we can also use the multiplication patterns a 1 b 2 5 a 2 1 2ab 1 b 2 a 2 b 2 5 a 2 2 2ab 1 b 2 a 1 b a 2 b 5 a 2 2 b 2 90360_ch0_001 102 indd 85 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 86 The last example illustrates an important idea The complex numbers 2 1 3i and 2 2 3i are called conjugates of each other In general th
132. ivities 90360_ch0_001 102 indd 64 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 6 Radicals 65 Thus 4 and 24 are both square roots of 16 In general a is a square root of b if a2 5 b The following statements generalize these ideas 1 Every positive real number has two square roots one is positive and the other is negative They are opposites of each other 2 Negative real numbers have no real number square roots because the square of any nonzero real number is positive 3 The square root of zero is zero The symbol 1 called a radical sign is used to designate the nonnegative square root which is called the principal square root The number under the radical sign is called the radicand and the entire expression such as 116 is referred to as a radical The following examples demonstrate the use of the square root notation 216 5 4 216 indicates the nonnegative or principal square root of 16 2216 5 24 2216 indicates the negative square root of 16
133. l numbers as sociated with points in the xy plane are of the form x y that is x is the rst coordinate and y is the second coordinate Graphing Utilities The term graphing utility is used in current literature to refer to either a graphing calculator see Figure 0 12 or a computer with a graphing software package We will frequently use the phrase use a graphing calculator to mean either a graphing calculator or a computer with an appropriate software package We will introduce various features of graphing calculators as we need them in the text Because so many different types of 90360_ch0_001 102 indd 14 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 15 Unless otherwise noted all art on this page is Cengage Learning graphing utilities are available we will use mostly generic terminology and let you con sult a user s manual for speci c key punching instructions We urge you to study the graphing calculator examples in this text even if
134. lex numbers be a real number Explain your answer Further investigations 88 Observe the following powers of i i 5 221 i2 5 21 i3 5 i2 i 5 21 i 5 2i i4 5 i2 i2 5 21 21 5 1 Any power of i greater than 4 can be simpli ed to i 21 2i or 1 as follows i9 5 i4 2 i 5 1 i 5 i i14 5 i4 3 i2 5 1 21 5 21 i19 5 i4 4 i3 5 1 2i 5 2i i28 5 i4 7 5 1 7 5 1 Express each of the following as i 21 2i or 1 a i 5 b i 6 c i 11 d i 12 e i 16 f i 22 g i 33 h i 63 Thoughts into Words 90360_ch0_001 102 indd 89 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 90 89 We can use the information from Problem 88 and the binomial expansion patterns to nd powers of complex numbers as follows 3 1 2i 3 5 3 3 1 3 3 2 2i 1 3 3 2i 2 1 2i 3 5 27 1 54i 1 36i2 1 8i3 5 27 1 54i 1 36 21 1 8 2i 5 29 1 46i Find the indicated power of each expression a 2 1 i 3
135. m of 7 is not produced by any of these pairs so the polynomial x 2 1 7x 1 16 is not factorable using integers Trinomials of the Form ax 2 1 bx 1 c Now let s consider factoring trinomials where the coefficient of the squared term is not one First let s illustrate an informal trial and error technique that works well for cer tain types of trinomials This technique is based on our knowledge of multiplication of binomials classroom example Factor a2 1 12a 1 32 classroom example Factor y2 2 10y 2 24 classroom example Factor x2 1 2x 1 12 90360_ch0_001 102 indd 44 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 4 Factoring Polynomials 45 eXAMPLe 4 Factor 3x 2 1 5x 1 2 solution By looking at the first term 3x 2 and the positive signs of the other two terms we know that the binomials are of the form x 1 __ 3x 1 __ Because the factors of the last term 2 are 1 and 2 we have only the following two possibilities to try x 1 2 3x
136. mbers b Nonpositive integers c Natural numbers Apply the definition of the absolute value of a number section 0 1 Objective 4 For all real numbers a If a 0 then 0a0 5 a If a 0 then 0a0 5 2a The following properties of ab solute value are useful 1 0a0 0 2 0a0 5 02a0 3 0a 2 b0 5 0b 2 a0 Evaluate 20x 2 y0 0y 2 x0 solution By the absolute value properties 0x 2 y0 5 0y 2 x Therefore 20x 2 y0 0y 2 x0 5 2 1 5 2 Know the real number properties section 0 1 Objective 5 As you study the operations on the set of real numbers the fol lowing properties will serve as the bases for many algebraic operations Commutative properties for addition and multiplication Associative properties for ad dition and multiplication Identity properties for addi tion and multiplication Inverse properties for addi tion and multiplication Distributive property State the property that justifies the statement a a 1 b 1 c 5 b 1 a 1 c b x 1 y 1 z 5 x 1 y 1 z c 4m 1 4n 5 4 m 1 n solution a Commutative property of addition b Associative property of addition c Distributive property 90360_ch0_001 102 indd 91 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial r
137. metric representation of the set of real numbers in front of us as indicated in Figure 0 2 Such a representation called the real number line indicates a one to one correspondence between the set of real numbers and the points on a line In other words to each real number there corresponds one and only one point on the line and to each point on the line there corresponds one and only one real number The number that corresponds to a particular point on the line is called the coordinate of that point 5 4 3 2 1 0 1 2 3 4 5 v2 1 2 1 2 v2 Figure 0 2 Many operations relations properties and concepts pertaining to real numbers can be given a geometric interpretation on the number line For example the addition prob lem 21 1 22 can be interpreted on the number line as shown in Figure 0 3 2 1 0 1 2 3 4 1 2 1 2 3 Figure 0 3 The inequality relations also have a geometric interpretation The statement a b read a is greater than b means that a is to the right of b and the statement c d read c is less than d means that c is to the left of d see Figure 0 4 The property 2 2x 5 x can be pictured on the number line in a sequence of steps See Figure 0 5 1 Choose a point that has a coordinate of x x 0 a x 0 b x 0 x c x 2 Locate its opposite written as 2x on the other s
138. n for a 1 b 6 a 1 b 6 5 a 6 1 6a 5b 1 15a 4b 2 1 20a 3b 3 1 15a 2b 4 1 6ab 5 1 b 6 Remark The triangular formation of numbers that we have been discussing is often referred to as Pascal s triangle This is in honor of Blaise Pascal a 17th century mathematician to whom the discovery of this pattern is attributed Let s consider two more examples using Pascal s triangle and the exponent rela tionships eXAMPLe 1 Expand a 2 b 4 solution We can treat a 2 b as a 2b and use the fourth row of Pascal s triangle 1 4 6 4 1 to obtain the coefficients a 1 2b 4 5 a 4 1 4a 3 2b 1 6a 2 2b 2 1 4a 2b 3 1 2b 4 5 a 4 2 4a 3b 1 6a 2b 2 2 4ab 3 1 b 4 eXAMPLe 2 Expand 2x 1 3y 5 solution Let 2x a and 3y b The coefficients 1 5 10 10 5 1 come from the fifth row of Pascal s triangle Dividing Polynomials by Monomials In Section 0 5 we will review the addition and subtraction of rational expressions using the properties a b 1 c b 5 a 1 c b and a b 2 c b 5 a 2 c b These properties can also be viewed as a 1 c b 5 a b 1 c b and a 2 c b 5 a b 2 c b classroom example Expand x 2 y 5 classroom example Expand 3a 1 2b 4 2x 1 3y 5 5 2x 5 1 5 2x 4 3y 1 10 2x 3 3y 2 1 10 2x 2 3y 3 1 5 2x 3y 4 1 3y 5 5 32x 5 1 240x 4y 1 720x 3y 2 1 1080x 2y 3 1 810xy 4 1 243y 5 90360_ch0_001 102 indd 37 11 17 11 8 47 AM Copyright 2012 Cengage Learning
139. n this section Objective 2 11 3xy 4x 2y 1 5xy 2 12 22ab 2 3a 2b 2 4ab 3 13 6a 3b 2 5ab 2 4a 2b 1 3ab 2 14 2xy 4 5x 2y 2 4xy 2 1 3x 2y 2 15 x 1 8 x 1 12 16 x 2 9 x 1 6 17 n 2 4 n 2 12 18 n 1 6 n 2 10 19 s 2 t x 1 y 20 a 1 b c 1 d 21 3x 2 1 2x 1 3 22 5x 1 2 3x 1 4 23 4x 2 3 3x 2 7 24 4n 1 3 6n 2 1 25 x 1 4 2 26 x 2 6 2 27 2n 1 3 2 28 3n 2 5 2 29 x 1 2 x 2 4 x 1 3 30 x 2 1 x 1 6 x 2 5 31 x 2 1 2x 1 3 3x 2 2 32 2x 1 5 x 2 4 3x 1 1 33 x 2 1 x 2 1 3x 2 4 34 t 1 1 t 2 2 2t 2 4 35 t 2 1 t 2 1 t 1 1 36 2x 2 1 x 2 1 4x 1 3 37 3x 1 2 2x 2 2 x 2 1 38 3x 2 2 2x 2 1 3x 1 4 39 x 2 1 2x 2 1 x 2 1 6x 1 4 40 x 2 2 x 1 4 2x 2 2 3x 2 1 41 5x 2 2 5x 1 2 42 3x 2 4 3x 1 4 43 x 2 2 5x 2 2 2 44 2x 2 1 x 2 1 2 45 2x 1 3y 2x 2 3y 46 9x 1 y 9x 2 y 47 x 1 5 3 48 x 2 6 3 49 2x 1 1 3 50 3x 1 4 3 51 4x 2 3 3 52 2x 2 5 3 53 5x 2 2y 3 54 x 1 3y 3 For Problems 55 66 use Pascal s triangle to help ex pand each expression Objective 3 55 a 1 b 7 56 a 1 b 8 57 x 2 y 5 58 x 2 y 6 59 x 1 2y 4 60 2x 1 y 5 61 2a 2 b 6 62 3a 2 b 4 63 x 2 1 y 7 64 x 1 2y 2 7 65 2a 2 3b 5 66 4a 2 3b 3 For Problems 67 72 perform the indicated divisions 67 15x 4 2 25x 3 5x 2 68 248x8 2 72x6 28x4 69
140. ns of equality and inequality the following properties will guide your study Be sure that you understand these properties because they not only facilitate manipulations with real numbers but also serve as a basis for many algebraic computations The variables a b and c represent real numbers Properties of Real Numbers Closure properties a 1 b is a unique real number ab is a unique real number Commutative properties a 1 b 5 b 1 a ab 5 ba Associative properties a 1 b 1 c 5 a 1 b 1 c ab c 5 a bc A B 2 1 0 1 2 3 4 Figure 0 7 90360_ch0_001 102 indd 8 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 9 Identity properties There exists a real number 0 such that a 1 0 5 0 1 a 5 a There exists a real number 1 such that a 1 5 1 a 5 a Inverse properties For every real number a there exists a unique real number 2a such that a 1 2a 5 2a
141. nteger6 Q 5 5x0x is a rational number6 H 5 5x0x is an irrational number6 R 5 5x0x is a real number6 Place or in each blank to make a true statement Objectives 1 and 2 19 N ________ R 20 R ________ N 21 N ________ I 22 I ________ Q 23 H ________ Q 24 Q ________ H 25 W ________ I 26 N ________ W 27 I ________ W 28 I ________ N 29 0 2 4 ________ W 30 1 3 5 7 ________ I 31 22 21 0 1 2 ________ W 32 0 3 6 9 ________ N For Problems 33 42 list the elements of each set For example the elements of x0 x is a natural number less than 4 can be listed 1 2 3 Objectives 1 and 2 33 x0 x is a natural number less than 2 34 x0 x is a natural number greater than 5 35 n0 n is a whole number less than 4 36 y0 y is an integer greater than 23 37 y0 y is an integer less than 2 38 n0 n is a positive integer greater than 24 39 x0 x is a whole number less than 0 40 x0 x is a negative integer greater than 25 41 n0 n is a nonnegative integer less than 3 42 n0 n is a nonpositive integer greater than 1 43 Find the distance on the real number line between two points whose coordinates are the following Objective 3 a 17 and 35 b 214 and 12 c 18 and 221 d 217 and 242 e 256 and 221 f 0 and 237 44 Evaluate each of the following if x is a nonzero real number Objective 4 a 0x0 x b x
142. o electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 36 Binomial expansion Pattern It is possible to write the expansion of a 1 b n where n is any positive integer with out showing all of the intermediate steps of multiplying and combining similar terms To do this let s observe some patterns in the following examples each one can be veri ed by direct multiplication a 1 b 1 5 a 1 b a 1 b 2 5 a 2 1 2ab 1 b 2 a 1 b 3 5 a 3 1 3a 2b 1 3ab 2 1 b 3 a 1 b 4 5 a 4 1 4a 3b 1 6a 2b 2 1 4ab 3 1 b 4 a 1 b 5 5 a 5 1 5a 4b 1 10a 3b 2 1 10a 2b 3 1 5ab 4 1 b 5 First note the patterns of the exponents for a and b on a term by term basis The expo nents of a begin with the exponent of the binomial and decrease by 1 term by term until the last term which has a0 5 1 The exponents of b begin with zero b0 5 1 and increase by 1 term by term until the last term which contains b to the power of the original bino mial In other words the variables in the expansion of a 1 b n have the pattern a n a n21b a n22b 2 ab n21 b n where for each term the sum of
143. of money in cents Algebra is often described as a generalized arithmetic That description does not tell the whole story but it does convey an important idea A good understanding of arithmetic provides a sound basis for the study of algebra In this chapter we will often use arithmetic examples to lead into a review of basic algebraic concepts Then we will use the algebraic concepts in a wide variety of problem solving situations Your study of algebra should make you a better problem solver Be sure that you can work effectively with the algebraic concepts reviewed in this rst chapter Some Basic Concepts of Algebra A Review 1 0 This statue of Fibonacci was constructed and erected in Pisa Italy Leonardo Fibonacci was a famous Italian middle ages mathematician He is known for spreading the Hindu Arabic number system in the western world and the Fibonacci sequence of numbers 0 1 Some Basic Ideas 0 2 Exponents 0 3 Polynomials 0 4 Factoring Polynomials 0 5 Rational Expressions 0 6 Radicals 0 7 Relationship Between Exponents and Roots 0 8 Complex Numbers David Lyons Alamy 90360_ch0_001 102 indd 1 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed cont
144. of the point from the horizontal axis measured parallel to the vertical axis Figure 0 10 Thus in the rst quadrant all points have a positive abscissa and a positive ordinate In the second quadrant all points have a negative abscissa and a positive ordinate We have indicated the sign situations for all four quadrants in Figure 0 11 This system of asso ciating points in a plane with pairs of real numbers is called the rectangular coordinate system or the Cartesian coordinate system a b a b Figure 0 10 Figure 0 11 Historically the rectangular coordinate system provided the basis for the develop ment of the branch of mathematics called analytic geometry or what we presently refer to as coordinate geometry In this discipline Ren Descartes a French 17th century mathematician was able to transform geometric problems into an algebraic setting and then use the tools of algebra to solve the problems Basically there are two kinds of problems to solve in coordinate geometry 1 Given an algebraic equation nd its geometric graph 2 Given a set of conditions pertaining to a geometric gure nd its algebraic equation Throughout this text we will consider a wide variety of situations dealing with both kinds of problems For most purposes in coordinate geometry it is customary to label the horizontal axis the x axis and the vertical axis the y axis Then ordered pairs of rea
145. of a The product of a number and its multiplicative inverse is the identity element for multiplication For example the reciprocal of 2 is 1 2 and 2 a1 2b 5 1 2 2 5 1 The product of any real number and zero is zero For example 217 0 5 0 217 5 0 The product of any real number and 21 is the opposite of the real num ber For example 21 52 5 52 21 5 252 The distributive property ties together the operations of addition and mul tiplication We say that multiplication distributes over addition For example 7 3 1 8 5 7 3 1 7 8 Furthermore because b 2 c 5 b 1 2c it follows that mul tiplication also distributes over subtraction This can be expressed symbolically as a b 2 c 5 ab 2 ac For example 6 8 2 10 5 6 8 2 6 10 Algebraic expressions Algebraic expressions such as 2x 8xy 23xy 24abc z are called terms A term is an indicated product and may have any number of factors The variables of a term are called literal factors and the numerical factor is called the numerical coef cient Thus in 8xy the x and y are literal factors and 8 is the numerical coef cient Because 1 z 5 z the numerical coef cient of the term z is understood to be 1 Terms that have the same literal factors are called similar terms or like terms The distributive property in the form ba 1 ca 5 b 1 c a provides the basis for simplifying algebraic expressions by combining similar terms as illus
146. ole number approximation Then use your calculator and the appropriate root keys to check your answers a 2 3 24 b 2 3 32 c 2 3 150 d 2 3 200 e 2 4 50 f 2 4 250 Answers to the concept Quiz 1 True 2 False 3 False 4 True 5 False 6 False 7 True 8 True 0 7 Relationship Between Exponents and Roots O B j e c T i V e s 1 Evaluate a number raised to a rational exponent 2 Simplify expressions with rational exponents 3 Apply rational exponents to simplify radical expressions Graphing calculator Activities 90360_ch0_001 102 indd 75 11 17 11 8 48 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 76 Recall that we used the basic properties of positive integral exponents to motivate a definition of negative integers as exponents In this section we shall use the properties of integral exponents to motivate definitions for rational numbers as exponents These definitions will tie together the concepts of exponent and roo
147. on the following definition a b 4 c d 5 a b d c 5 ad bc Find the quotient 6xy x2 2 6x 1 9 4 18x x2 2 9 solution 6xy x2 2 6x 1 9 4 18x x2 2 9 5 6xy x2 2 6x 1 9 x2 2 9 18x 5 6xy x 2 3 x 2 3 x 1 3 x 2 3 18x 6xy x 2 3 x 2 3 x 1 3 x 2 3 18x 3 5 y x 1 3 3 x 2 3 Add and subtract rational expressions section 0 5 Objective 3 Addition and subtraction of ra tional expressions are based on the following definitions a b 1 c b 5 a 1 c b Addition a b 2 c b 5 a 2 c b Subtraction The following basic procedure is used to add or subtract rational expressions 1 Factor the denominators 2 Find the LCD 3 Change each fraction to an equivalent fraction that has the LCD as the denominator 4 Combine the numerators and place over the LCD 5 Simplify by performing the addition or subtraction in the numerator 6 If possible reduce the result ing fraction Subtract 2 x2 2 2x 2 3 2 5 x2 1 5x 1 4 solution 2 x2 2 2x 2 3 2 5 x2 1 5x 1 4 5 2 x 2 3 x 1 1 2 5 x 1 1 x 1 4 The LCD is x 2 3 x 1 1 x 1 4 5 2 x 1 4 x 2 3 x 1 1 x 1 4 2 5 x 2 3 x 1 1 x 1 4 x 2 3 5 2 x 1 4 2 5 x 2 3 x 2 3 x 1 1 x 1 4 5 2x 1 8 2 5x 1 15 x 2 3 x 1 1 x 1 4 5 23x 1 23 x 2 3 x 1 1 x 1 4 OBjecTiVe sUMMARy eXAMPLe 90360_ch0_001 102 indd 96 11 17
148. oncept Quiz 1 True 2 False 3 True 4 True 5 False 6 False 7 False 8 False Thoughts into Words Graphing calculator Activities 90360_ch0_001 102 indd 40 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 4 Factoring Polynomials 41 If a polynomial is equal to the product of other polynomials then each polynomial in the product is called a factor of the original polynomial For example because x2 2 4 can be expressed as x 1 2 x 2 2 we say that x 1 2 and x 2 2 are factors of x2 2 4 The process of expressing a polynomial as a product of polynomials is called factoring In this section we will consider methods of factoring polynomials with integer coefficients In general factoring is the reverse of multiplication so we can use our knowledge of multiplication to help develop factoring techniques For example we pre viously used the distributive property to find the product of a monomial and a poly nomial as the next examples illustrate 3 x 1 2
149. pressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Answers to Odd Numbered Problems A 3 79 x2 1 x 1 1 x 1 1 81 a2 1 4a 1 1 4a 1 1 83 2 2x 1 h x2 x 1 h 2 85 2 1 x 1 1 x 1 h 1 1 87 2 4 2x 2 1 2x 1 2h 2 1 89 y 1 2x x2y 2 xy2 91 x2y2 1 2 4y2 2 3x Problem Set 0 6 page 73 1 9 3 5 5 6 7 7 23 2 9 226 11 427 13 26211 15 325 2 17 2x23 19 8x2y32y 21 9y325x 7 23 42 3 2 25 2x2 3 2x 27 2x2 4 3x 29 223 5 31 214 4 33 4215 5 35 322 7 37 215 6x2 39 2215a 5ab 41 32 3 2 2 43 2 3 18x2y 3x 45 1223 47 327 49 1123 6 51 28922 30 53 4826
150. r as follows 5x 2 2 18x 2 8 5 5x 2 2 20x 1 2x 2 8 5 5x x 2 4 1 2 x 2 4 5 x 2 4 5x 1 2 eXAMPLe 10 Factor 24x 2 1 2x 2 15 solution 24x 2 1 2x 2 15 Sum of 2 Product of 24 215 5 2360 We need two integers whose sum is 2 and whose product is 2360 To help find these integers let s factor 360 into primes 360 5 2 2 2 3 3 5 Now by grouping these factors in various ways we find that 2 2 5 5 20 and 2 3 3 5 18 so we can use the integers 20 and 218 to produce a sum of 2 and a product of 2360 Therefore the middle term 2x of the trinomial can be expressed as 20x 2 18x and we can proceed as follows 24x 2 1 2x 2 15 5 24x 2 1 20x 2 18x 2 15 5 4x 6x 1 5 2 3 6x 1 5 5 6x 1 5 4x 23 Probably the best way to check a factoring problem is to make sure the conditions for a polynomial to be completely factored are satisfied and the product of the factors classroom example Factor 3y2 1 16y 2 12 classroom example Factor 8a2 1 22a 2 21 90360_ch0_001 102 indd 47 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppress
151. r different values of x Let x 5 3 then 2x2 5 232 5 19 5 3 Let x 5 23 then 2x2 5 2 23 2 5 19 5 3 Thus if x 0 then 2x2 5 x but if x 0 then 2x2 5 2x Using the concept of absolute value we can state that for all real numbers 2x2 x Now consider the radical 2x3 Because x 3 is negative when x is negative we need to restrict x to the nonnegative real numbers when working with 2x3 Thus we can write if x 0 then 2x3 5 2x21x 5 x1x and no absolute value sign is needed Finally let s consider the radical 2 3 x3 Let x 5 2 then 2 3 x3 5 2 3 23 5 2 3 8 5 2 Let x 5 22 then 2 3 x3 5 2 3 22 3 5 2 3 28 5 22 Thus it is correct to write 2 3 x3 5 x for all real numbers and again no absolute value sign is needed The previous discussion indicates that technically every radical expression with variables in the radicand needs to be analyzed individually to determine the necessary restrictions on the variables However to avoid having to do this on a problem by problem basis we shall merely assume that all variables represent positive real numbers Let s conclude this section by simplifying some radical expressions that contain variables 272x2y7 5 236x2y622xy 5 6xy322xy 2 3 40x4y8 5 2 3 8x3y6 2 3 5xy2 5 2xy2 2 3 5xy2 25 212a3 5 25 212a3 23a 23a 5 215a 236a4 5 215a 6a2 3 2 3 4x 5 3 2 3 4x 2 3 2x2 2 3 2x2 5 32 3
152. rd party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 34 Unless otherwise noted all art on this page is Cengage Learning In this example we are claiming that x 2 2 x 2 2 3x 1 4 5 x 3 2 5x 2 1 10x 2 8 for all real numbers In addition to going back over our work how can we verify such a claim Obviously we cannot try all real numbers but trying at least one number gives us a partial check Let s try the number 4 x 2 2 x 2 2 3x 1 4 5 4 2 2 4 2 2 3 4 1 4 5 2 16 2 12 1 4 5 2 8 5 16 x 3 2 5x 2 1 10x 2 8 5 4 3 2 5 4 2 1 10 4 2 8 5 64 2 80 1 40 2 8 5 16 We can also use a graphical approach as a partial check for such a problem In Figure 0 18 we let Y1 5 x 2 2 x 2 2 3x 1 4 and Y2 5 x 3 2 5x 2 1 10x 2 8 and graphed them on the same set of axes Note that the graphs appear to be identical 15 215 10 210 Figure 0 18 Remark Graphing on the Cartesian coordinate system is not formally reviewed in this text until Chapter 2 However we feel confident that your knowledge of this topic from previous mathematics courses is sufficient for what w
153. right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 74 94 2288x 6 95 228m 8 96 2128c10 97 218d 7 98 249x 2 99 280n 20 100 281h 3 Do the following problems where the variable could be any real number as long as the radical represents a real number Use absolute value signs in the answers as necessary 90 2125x2 91 216x4 92 28b3 93 23y5 71 4 27 2 23 72 2 25 1 23 73 22 225 1 327 74 5 522 2 325 75 1x 1x 2 1 76 1x 1x 1 2 77 1x 1x 1 1y 78 21x 1x 2 1y 79 21x 1 1y 31x 2 21y 80 31x 2 21y 21x 1 51y For Problems 81 84 rationalize the numerator All variables represent positive real numbers Objective 5 81 22x 1 2h 2 22x h 82 2x 1 h 1 1 2 2x 1 1 h 83 2x 1 h 2 3 2 2x 2 3 h 84 22x 1 h 2 22x h For Problems 53 68 multiply and express the results in simplest radical form All variables represent non negative real numbers Objective 4 53 A413BA618B 54 A518BA317B 55 213A512 1 4110B 56 316A218 2 3112B 57 32xA 26xy 2 28yB 58 26yA 28x 1 210y2B 59 A 23 1 2BA 23 1 5B 60 A 22 2 3BA 22 1 4B 61 A422 1 23BA322 1 223B 62 A226 1 325BA326 1 425B 63 A6 1 225BA62225B 64 A7 2 322BA7 1 322B 65 A 2x 1 2yB2 66 A21x 2 31yB2 67 A 1a 1 2bBA 1a 2 2bB 68 A31x 1 51yBA31x 2
154. roperty in the form ab 1 ac 5 a b 1 c is the basis for factoring out a common factor The common factor can be a bi nomial factor as when perform ing factoring by grouping Factor 26x5y4 2 3x6y3 2 24x7y2 solution The common factor is 23x5y2 26x5y4 2 3x6y3 2 24x7y2 5 23x5y2 2y2 1 xy 1 8x2 Factor by grouping section 0 4 Objective 2 It may be that the polynomial exhibits no common monomial or binomial factor However by factoring common factors from groups of terms a common fac tor may be evident Factor 2xz 1 6x 1 yz 1 3y solution 2xz 1 6x 1 yz 1 3y 5 2x z 1 3 1 y z 1 3 5 z 1 3 2x 1 y OBjecTiVe sUMMARy eXAMPLe Factor the difference of two squares section 0 4 Objective 3 The factoring pattern a2 2 b2 5 a 1 b a 2 b is called the difference of two squares Factor 36a2 2 25b2 solution 36a2 2 25b2 5 6a 2 5b 6a 1 5b 90360_ch0_001 102 indd 94 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 95 Factor trinomials of the form
155. s x 1 3 2 2 y 2 5 x 1 3 1 y x 1 3 2 y 5 x 1 3 1 y x 1 3 2 y 4x 2 2 2y 1 1 2 5 2x 1 2y 1 1 2x 2 2y 1 1 5 2x 1 2y 1 1 2x 2 2y 2 1 x 2 1 2 2 x 1 4 2 5 x 2 1 1 x 1 4 x 2 1 2 x 1 4 5 x 2 1 1 x 1 4 x 2 1 2 x 2 4 5 2x 1 3 25 It is possible that both the technique of factoring out a common monomial factor and the pattern of the difference of two squares can be applied to the same problem In general it is best to look first for a common monomial factor Consider the following examples 2x 2 2 50 5 2 x 2 2 25 5 2 x 1 5 x 2 5 48y 3 2 27y 5 3y 16y 2 2 9 5 3y 4y 1 3 4y 2 3 9x 2 2 36 5 9 x 2 2 4 5 9 x 1 2 x 2 2 90360_ch0_001 102 indd 43 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User
156. s 83 The set of real numbers is a subset of the set of complex numbers The following diagram indicates the organizational format of the complex number system Complex numbers a 1 bi where a and b are real numbers Real numbers Imaginary numbers a 1 bi where b 5 0 a 1 bi where b 0 Pure imaginary numbers a 1 bi where a 5 0 and b 0 Two complex numbers a 1 bi and c 1 di are said to be equal if and only if a 5 c and b 5 d In other words two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal Adding and subtracting complex Numbers The following definition provides the basis for adding complex numbers a 1 bi 1 c 1 di 5 a 1 c 1 b 1 d i We can use this definition to find the sum of two complex numbers 4 1 3i 1 5 1 9i 5 4 1 5 1 3 1 9 i 5 9 1 12i 26 1 4i 1 8 2 7i 5 26 1 8 1 4 2 7 i 5 2 2 3i a1 2 1 3 4 ib 1 a2 3 1 1 5 ib 5 a1 2 1 2 3b 1 a3 4 1 1 5b i 5 a3 6 1 4 6b 1 a15 20 1 4 20b i 5 7 6 1 19 20 i A3 1 i22B 1 A2 4 1 i22B 5 3 1 2 4 1 A 22 1 22Bi 5 2 1 1 2i22 Note the form for writing 222i The set of complex numbers is closed with respect to addition that is the sum of two complex numbers is a complex number Furthermor
157. s The degree of a monomial is the sum of the exponents of the literal factors For example 7xy is of degree 2 whereas 14a 2b is of degree 3 and 217ab 2c 3 is of degree 6 If the monomial contains only one variable then the exponent of that variable is the degree of the monomial For example 5x 3 is of degree 3 and 28y 4 is of degree 4 Any nonzero constant term such as 8 is of degree zero A polynomial is a monomial or a nite sum of monomials Thus all of the follow ing are examples of polynomials 4x 2 3x 2 2 2x 2 4 7x 4 2 6x 3 1 5x 2 2 2x 2 1 3x 2y 1 2y 1 5 a2 2 2 3 b2 14 In addition to calling a polynomial with one term a monomial we classify poly nomials with two terms as binomials and those with three terms as trinomials The degree of a polynomial is the degree of the term with the highest degree in the polynomial The following examples illustrate some of this terminology The polynomial 4x 3y 4 is a monomial in two variables of degree 7 The polynomial 4x 2y 2 2xy is a binomial in two variables of degree 3 The polynomial 9x 2 2 7x 2 1 is a trinomial in one variable of degree 2 90360_ch0_001 102 indd 31 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not ma
158. scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 88 Some of the solution sets for quadratic equations in the next chapter will contain complex numbers such as 24 1 2212 2 and 24 2 2212 2 We can simplify the first number as follows 24 1 2212 2 5 24 1 i212 2 5 24 1 2i23 2 5 2 22 1 i23 2 5 22 1 i23 For Problems 31 36 simplify each of the following complex numbers 31 24 2 2212 2 32 6 1 2224 4 33 23 2 2218 3 34 26 1 2227 3 35 12 1 2245 6 36 4 2 2248 2 For Problems 37 50 write each in terms of i perform the indicated operations and simplify Objective 1 For example 2292216 5 Ai29BAi216B 5 3i 4i 5 12i2 5 12 21 5 212 37 2242216 38 2225229 39 222223 40 223227 41 225224 42 227229 43 2262210 44 2222212 45 228227 46 2212225 47 2236 224 48 2264 2216 49 2254 229 50 2218 223 For Problems 1 14 add or subtract as indicated Objective 2 1 5 1 2i 1 8 1 6i 2 29
159. scientific form section 0 2 Objective 5 Scientific notation can be used to simplify numerical operations by changing the numbers to scientific notation and using the appropriate properties of exponents Simplify 0 0000068 0 04 solution 0 0000068 0 04 5 6 8 1026 4 1022 5 1 7 1024 5 0 00017 Add and subtract polynomials section 0 3 Objective 1 Similar or like terms have the same literal factors The commu tative associative and distribu tive properties provide the basis for rearranging regrouping and combining similar terms Simplify 5x 2 3x2 2 4 6x 2 2x2 solution 5x 2 3x2 2 4 6x 2 2x2 5 5x 2 3x2 2 24x 1 8x2 5 5x 2 11x2 2 24x 5 5x 2 11x2 1 24x 5 211x2 1 29x Multiply polynomials section 0 3 Objective 2 To multiply two polynomials every term of the first polyno mial is multiplied by each term of the second polynomial Multiplying polynomials often produces similar terms that can be combined to simplify the re sulting polynomial Find the indicated product 3x 1 5 x2 2 2x 1 7 solution 3x 1 5 x2 2 2x 1 7 5 3x x2 2 2x 1 7 1 5 x2 2 2x 1 7 5 3x3 2 6x2 1 21x 1 5x2 2 10x 1 35 5 3x3 2 x2 1 11x 1 35 OBjecTiVe sUMMARy eXAMPLe Perform binomial expansions section 0 3 Objective 3 It is possible to write the expan sion of a 1 b n where n is a natural number without doing all the intermediate steps This can be done by realizing
160. see that the LCD is 12 x 1 2 4 1 3x 1 1 3 5 ax 1 2 4 ba3 3b 1 a3x 1 1 3 ba4 4b 5 3 x 1 2 12 1 4 3x 1 1 12 5 3x 1 6 1 12x 1 4 12 5 15x 1 10 12 eXAMPLe 2 Perform the indicated operations x 1 3 10 1 2x 1 1 15 2 x 2 2 18 solution If you cannot determine the LCD by inspection then use the prime factored forms of the denominators 10 5 2 5 15 5 3 5 18 5 2 3 3 The LCD must contain one factor of 2 two factors of 3 and one factor of 5 Thus the LCD is 2 3 3 5 5 90 x 1 3 10 1 2x 1 1 15 2 x 2 2 18 5 ax 1 3 10 ba9 9b 1 a2x 1 1 15 ba6 6b 2 ax 2 2 18 ba5 5b 5 9 x 1 3 90 1 6 2x 1 1 90 2 5 x 2 2 90 5 9x 1 27 1 12x 1 6 2 5x 1 10 90 5 16x 1 43 90 classroom example Add 3x 1 2 5 1 x 1 6 4 classroom example Perform the indicated operations x 1 7 20 1 3x 2 4 12 2 x 2 3 18 90360_ch0_001 102 indd 55 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require i
161. sequent rights restrictions require it Licensed to CengageBrain User 0 6 Radicals 69 Property 0 4 states that the nth root of a quotient is equal to the quotient of the nth roots To evaluate radicals such as B 4 25 and B 3 27 8 where the numerator and the de nominator of the fractional radicands are perfect nth powers we can either use Property 0 4 or rely on the definition of nth root B 4 25 5 24 225 5 2 5 or B 4 25 5 2 5 because 2 5 2 5 5 4 25 B 3 27 8 5 2 3 27 2 3 8 5 3 2 or B 3 27 8 5 3 2 because 3 2 3 2 3 2 5 27 8 Radicals such as B 28 9 and B 3 24 27 where only the denominators of the radi cand are perfect nth powers can be simplified as follows B 28 9 5 228 29 5 2427 3 5 227 3 B 3 24 27 5 2 3 24 2 3 27 5 2 3 82 3 3 3 5 22 3 3 3 Before we consider more examples let s summarize some ideas about simplifying radicals A radical is said to be in simplest radical form if the following conditions are satisfied 1 No fraction appears within a Thus B 3 4 violates this condition radical sign 2 No radical appears in the Thus 22 23 violates this condition denominator 3 No radicand contains a perfect Thus 272 5 violates this power of the index condition Rationalizing Radical expressions Now let s consider an example in whic
162. t Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 56 The presence of variables in the denominators does not create any serious diffi culty our approach remains the same Study the following examples very carefully For each problem we use the same basic procedure 1 Find the LCD 2 Change each fraction to an equivalent fraction having the LCD as its denominator 3 Add or sub tract numerators and place this result over the LCD 4 Look for possibilities to sim plify the resulting fraction eXAMPLe 3 Add 3 2x 1 5 3y solution Using an LCD of 6xy we can proceed as follows 3 2x 1 5 3y 5 a 3 2xba3y 3yb 1 a 5 3yba2x 2xb 5 9y 6xy 1 10x 6xy 5 9y 1 10x 6xy eXAMPLe 4 Subtract 7 12ab 2 11 15a2 solution We can factor the numerical coefficients of the denominators into primes to help find the LCD 12ab 5 2 2 3 a b 15a2 5 3 5 a2 f LCD 5 2 2 3 5 a2 b 5 60a2b 7 12ab 2 11 15a2 5 a 7 12abba5a 5ab 2 a 11 15a2ba4b 4bb 5 35a 60a2b 2 44b 60a2b 5 35a 2 44b 60a2b eXAMPLe 5 Add 8 x2 2 4x 1 2 x solution x2 2 4x 5 x x 2 4 x 5 x f LCD 5 x x 2 4 classroom example Add 2 3a 1 4 5b classroom example Subtract 1 6x2 2 5 9xy classroom example Add 3 y 1 6 y2 2 2y 90360_ch0_001 102 indd 56 11 17 11 8 47 AM Copyri
163. t Let s consider the fol lowing comparisons From our study of If b m n 5 b nm is to hold when m is a rational radicals we know number of the form 1 gt p where p is a positive that integer greater than 1 and n 5 p then A 25B2 5 5 51 gt 2 2 5 5 211 gt 22 5 5 1 5 5 A 2 3 8B3 5 8 81 gt 3 3 5 8 311 gt 32 5 8 1 5 8 A 2 4 21B4 5 21 211 gt 4 4 5 21 411 gt 42 5 211 5 21 Such examples motivate the following definition Definition 0 6 If b is a real number n is a positive integer greater than 1 and 1 nb exists then b1 n 5 2 n b Definition 0 6 states that b 1 gt n means the nth root of b We shall assume that b and n are chosen so that 2 n b exists in the real number system For example 225 1y2 is not meaningful at this time because 2225 is not a real number The following examples illustrate the use of Definition 0 6 251y2 5 225 5 5 16 1 gt 4 5 2 4 16 5 2 81y3 5 2 3 8 5 2 227 1y3 5 2 3 227 5 23 Now the following definition provides the basis for the use of all rational numbers as exponents Definition 0 7 If m n is a rational number expressed in lowest terms where n is a positive in teger greater than 1 and m is any integer and if b is a real number such that 1 nb exists then b m gt n 5 2 n bm 5 A 2 n b Bm In Definition 0 7 whether we use the form 2 n bm or A 2 n bBm for computational purposes depends somewhat on the ma
164. t example notice that each term of the rst polynomial multiplies each term of the second polynomial x 2 3 y 1 z 1 3 5 x y 1 z 1 3 2 3 y 1 z 1 3 5 xy 1 xz 1 3x 2 3y 2 3z 2 9 Frequently multiplying polynomials produces similar terms that can be combined which simpli es the resulting polynomial x 1 5 x 1 7 5 x x 1 7 1 5 x 1 7 5 x 2 1 7x 1 5x 1 35 5 x 2 1 12x 1 35 In a previous algebra course you may have developed a shortcut for multiplying bino mials as illustrated by Figure 0 17 2x 5 3x 2 6x2 11x 10 2 1 3 Figure 0 17 sTeP 1 Multiply 2x 3x STEP 2 Multiply 5 3x and 2x 22 and combine STEP 3 Multiply 5 22 Remark Shortcuts can be very helpful for certain manipulations in mathe matics But a word of caution Do not lose the understanding of what you are doing Make sure that you are able to do the manipulation without the shortcut Keep in mind that the shortcut illustrated in Figure 0 17 applies only to multiplying two binomials The next example applies the distributive property to find the product of a binomial and a trinomial x 2 2 x 2 2 3x 1 4 5 x x 2 2 3x 1 4 2 2 x 2 2 3x 1 4 5 x 3 2 3x 2 1 4x 2 2x 2 1 6x 2 8 5 x 3 2 5x 2 1 10x 2 8 90360_ch0_001 102 indd 33 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some thi
165. t of this correspon dence is illustrated in Figure 0 9 For example the ordered pair 3 2 means that the point A is located 3 units to the right of and 2 units up from the origin Likewise the ordered pair 23 25 means that the point D is located 3 units to the left of and 5 units down from the origin The ordered pair 0 0 is associated with the origin O O 0 0 A 3 2 E 5 2 B 2 4 C 4 0 D 3 5 Figure 0 9 In general we refer to the real numbers a and b in an ordered pair a b associated with a point as the coordinates of the point The rst number a called the abscissa is the directed distance of the point from the vertical axis measured parallel to the 90360_ch0_001 102 indd 13 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 14 Unless otherwise noted all art on this page is Cengage Learning horizontal axis The second number b called the ordinate is the directed distance
166. t rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 12 Look back at Examples 1 3 and note that we use the following order of operations when simplifying numerical expressions 1 Perform the operations inside the symbols of inclusion parentheses brackets and braces and above and below each fraction bar Start with the innermost inclusion symbol 2 Perform all multiplications and divisions in the order in which they appear from left to right 3 Perform all additions and subtractions in the order in which they appear from left to right You should also realize that rst simplifying by combining similar terms can some times aid in the process of evaluating algebraic expressions The last example of this section illustrates this idea eXAMPLe 4 Evaluate 2 3x 1 1 2 3 4x 2 3 when x 5 25 solution 2 3x 1 1 2 3 4x 2 3 5 2 3x 1 2 1 2 3 4x 2 3 23 2 3x 1 1 2 3 4x 2 3 5 6x 1 2 2 12x 1 9 2 3x 1 1 2 3 4x 2 3 5 26x 1 11 Now substituting 25 for x we obtain 26x 1 11 5 26 25 1 11 26x 1 11 5 30 1 11 26x 1 11 5 41 cartesian coordinate system Just as real numbers can be associated with points on a line pairs of real numbers can be associated with points in a plane To do this we set up two number lines one verti cal and one horizontal perpendicular to each other at the point associated with zero on both lines as shown in Figure 0 8 We ref
167. ted in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Answers to Odd Numbered Problems A 4 31 x4 1 2x3 2 6x2 2 22x 2 15 32 2x4 1 11x3 2 16x2 2 8x 1 8 33 24x2y3 1 8xy2 34 27y 1 9xy2 35 3x 1 2y 3x 2 2y 36 3x x 1 5 x 2 8 37 2x 1 5 2 38 x 2 y 1 3 x 2 y 2 3 39 x 2 2 x 2 y 40 4x 2 3y 16x2 1 12xy 1 9y2 41 3x 2 4 5x 1 2 42 3 x3 1 12 43 Not factorable 44 3 x 1 2 x2 2 2x 1 4 45 x 1 3 x 2 3 x 1 2 x 2 2 46 2x 2 1 2 y 2x 2 1 1 y 47 2 3y 48 25a2 3 49 3x 1 5 x 50 2 3x 2 1 x2 1 4 51 29x 2 10 12 52 x 2 38 15 53 26n 1 15 5n2 54 23x 2 16 x x 1 7 55 3x2 2 8x 2 40 x 1 4 x 2 4 x 2 10 56 8x 2 4 x x 1 2 x 2 2 57 3xy 2 2x2 5y 1 7x2
168. ter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 8 Unless otherwise noted all art on this page is Cengage Learning indicate that the absolute value of any real number is equal to the absolute value of its opposite All of these ideas are summarized in the following properties Properties of Absolute Value The variables a and b represent any real number 1 0 a 0 0 The absolute value of a real number is positive or zero 2 0 a 0 5 02a 0 The absolute value of a real number is equal to the absolute value of its opposite 3 0 a 2 b0 5 0 b 2 a 0 The expressions a 2 b and b 2 a are opposites of each other hence their absolute values are equal In Figure 0 7 the points A and B are located at 22 and 4 respectively The distance between A and B is 6 units and can be calculated by using either 022 2 4 0 or 04 2 22 0 In general if two points on a number line have coordinates x1 and x2 then the distance between the two points is determined by using either 0x2 2 x1 0 or 0x1 2 x2 0 because by property 3 above they are the same quantity Properties of Real Numbers As you work with the set of real numbers the basic operations and the relatio
169. terially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 32 Addition and subtraction of Polynomials Both adding polynomials and subtracting them rely on the same basic ideas The com mutative associative and distributive properties provide the basis for re arranging re grouping and combining similar terms Consider the following addition problems 4x 2 1 5x 1 1 1 7x 2 2 9x 1 4 5 4x 2 1 7x 2 1 5x 2 9x 1 1 1 4 5 11x 2 2 4x 1 5 5x 2 3 1 3x 1 2 1 8x 1 6 5 5x 1 3x 1 8x 1 23 1 2 1 6 5 16x 1 5 The de nition of subtraction as adding the opposite a 2 b 5 a 1 2b extends to polynomials in general The opposite of a polynomial can be formed by taking the opposite of each term For example the opposite of 3x 2 2 7x 1 1 is 23x 2 1 7x 2 1 Symbolically this is expressed as 2 3x 2 2 7x 1 1 5 23x 2 1 7x 2 1 You can also think in terms of the property 2x 5 21 x and the distributive property Therefore 2 3x 2 2 7x 1 1 5 21 3x 2 2 7x 1 1 5 23x 2 1 7x 2 1 Now consider the following subtraction problems 7x 2 2 2x 2 4 2 3x 2 1 7x 2 1 5 7x 2 2 2x 2 4 1 23x 2 2 7x 1 1 5 7x 2 2 3x 2 1 22x 2 7x 1 24 1 1 5 4x 2 2 9x 2 3 4y 2 1 7 2 23y 2 1 y 2 2 5 4y 2 1 7 1 3y 2 2 y 1 2
170. th the irrationals The following tree diagram can be used to summarize the various classi cations of the real number system 90360_ch0_001 102 indd 4 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 5 Unless otherwise noted all art on this page is Cengage Learning Any real number can be traced down through the tree Here are some examples 7 is real rational an integer and positive 22 3 is real rational a noninteger and negative 17 is real irrational and positive 0 59 is real rational a noninteger and positive The concept of a subset is convenient to use at this time A set A is a subset of another set B if and only if every element of A is also an element of B For example if A 5 1 2 and B 5 1 2 3 then A is a subset of B This is written A B and is read A is a subset of B The slash mark can also be used here to denote negation If A 5 1 2 4 6 and B 5 2 3 7 we can say A is not a subset of B by writing A
171. the pat tern of the exponents for each term of the expansion and using Pascal s triangle to determine the coefficient for each term Expand 2x 1 y 4 solution 2x 1 y 4 5 2x 4 1 4 2x 3y 1 6 2x 2y2 1 4 2x y3 1 y4 5 16x4 1 32x3y 1 24x2y2 1 8xy3 1 y4 continued Chapter 0 Summary 90360_ch0_001 102 indd 93 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 94 Divide a polynomial by a monomial section 0 3 Objective 4 To divide a polynomial by a monomial divide each term of the polynomial by the monomial Perform the indicated division 15a3b4 2 30a5b7 1 5a2b3 5a2b3 solution Rewrite the problem as separate fractions ob tained by each term in the numerator divided by the denominator Then simplify each fraction 15a3b4 2 30a5b7 1 5a2b3 5a2b3 5 15a3b4 5a2b3 2 30a5b7 5a2b3 1 5a2b3 5a2b3 5 3ab 2 6a3b4 1 1 Factor out a common factor section 0 4 Objective 1 The distributive p
172. trated in the following examples 3x 1 5x 5 3 1 5 x 5 8x 26xy 1 4xy 5 26 1 4 xy 5 22xy 4x 2 x 5 4x 2 1x 5 4 2 1 x 5 3x Sometimes we can simplify an algebraic expression by applying the distributive property to remove parentheses and combine similar terms as the next ex amples il lustrate 4 x 1 2 1 3 x 1 6 5 4 x 1 4 2 1 3 x 1 3 6 4 x 1 2 1 3 x 1 6 5 4x 1 8 1 3x 1 18 4 x 1 2 1 3 x 1 6 5 7x 1 26 25 y 1 3 2 2 y 2 8 5 25 y 2 5 3 2 2 y 2 2 28 25 y 1 3 2 2 y 2 8 5 25y 2 15 2 2y 1 16 25 y 1 3 2 2 y 2 8 5 27y 1 1 90360_ch0_001 102 indd 10 11 17 11 8 47 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User 0 1 Some Basic Ideas 11 An algebraic expression takes on a numerical value whenever each variable in the expression is replaced by a real number For example when x is replaced by 5 and y by 9 the algebraic expression x 1 y becomes the numerical expression 5 1 9 which is equal to 14 We say that x 1 y has a value of 14 when x 5 5 and y 5 9 Consider the following examples whic
173. ur answers for Problems 29 40 87 Use the graphing feature of your graphing calculator to give visual support for your answers for Problems 47 52 88 Some of the product patterns can be used to do arith metic computations mentally For example let s use the pattern a 1 b 2 5 a 2 1 2ab 1 b2 to com pute 312 mentally Your thought process should be 312 5 30 1 1 2 5 302 1 2 30 1 1 12 5 961 Compute each of the following numbers mentally and then check your answers with your calculator a 212 b 412 c 712 d 322 e 522 f 822 89 Use the pattern a 2 b 2 5 a 2 2 2ab 1 b2 to com pute each of the following numbers mentally and then check your answers with your calculator a 192 b 292 c 492 d 792 e 382 f 582 90 Every whole number with a units digit of 5 can be represented by the expression 10x 1 5 where x is a whole number For example 35 5 10 3 1 5 and 145 5 10 14 1 5 Now let s observe the following pattern when squaring such a number 10x 1 5 2 5 100x2 1 100x 1 25 5 100x x 1 1 1 25 The pattern inside the dashed box can be stated as add 25 to the product of x x 1 1 and 100 Thus to compute 352 mentally we can think 352 5 3 4 100 1 25 5 1225 Compute each of the fol lowing numbers mentally and then check your answers with your calculator a 152 b 252 c 452 d 552 e 652 f 752 g 852 h 952 i 1052 Answers to the c
174. verall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 16 Unless otherwise noted all art on this page is Cengage Learning Another method is shown in Figure 0 14 in which the values for x y and z are stored and then the algebraic expression 3xy 2 4z is evaluated Figure 0 14 For Problems 1 10 identify each statement as true or false Objective 2 1 Every rational number is a real number 2 Every irrational number is a real number 3 Every real number is a rational number 4 If a number is real then it is irrational Problem set 0 1 5 Some irrational numbers are also rational numbers 6 All integers are rational numbers 7 The number zero is a rational number 8 Zero is a positive integer 9 Zero is a negative number 10 All whole numbers are integers For Problems 1 10 answer true or false 1 The null set is written as 2 The sets a b c d and a d c b are equal sets 3 Decimal numbers that are classified as repeating or terminating decimals repre sent rational numbers 4 The absolute value of x is equal to x 5 The axes of the rectangular coordinate system intersect in a point called the center 6 Subtraction is a commutative operation 7 Every real number has a multiplicative invers
175. x2 1 bx 1 c and tri nomials of the form ax2 1 bx 1 c section 0 4 Objective 4 Expressing a trinomial for which the coefficient of the squared term is 1 as a product of two binomi als is based on the relationship x 1 a x 1 b 5 x2 1 a 1 b x 1 ab The coefficient of the middle term is the sum of a and b and the last term is the product of a and b Two methods were presented for factoring trinomials of the form ax2 1 bx 1 c One technique is to try the various possibilities of factors and check by multiplying This method is referred to as trial and error The other method is structured technique and is shown in Section 0 4 Examples 8 and 9 Factor x2 2 2x 2 35 solution x2 2 2x 2 35 5 x 2 7 x 1 5 Factor 4x2 1 16x 1 15 solution Multiply 4 times 15 to get 60 The factors of 60 that add to 16 are 6 and 10 Rewrite the problem and factor by grouping 4x2 1 16x 1 15 5 4x2 1 10x 1 6x 1 15 5 2x 2x 1 5 1 3 2x 1 5 5 2x 1 5 2x 1 3 Factor the sum or differ ence of two cubes section 0 4 Objective 5 The factoring patterns a3 1 b3 5 a 1 b a2 2 ab 1 b2 and a3 2 b3 5 a 2 b a2 1 ab 1 b2 are called the sum of two cubes or the difference of two cubes Factor 8x3 1 27y3 solution 8x3 1 27y3 5 2x 1 3y 4x2 2 6xy 1 9y2 Apply more than one fac toring technique section 0 4 Objective 6 Be sure to factor completely Some problems require that more than one factoring tech nique may be n
176. y suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Licensed to CengageBrain User Chapter 0 Some Basic Concepts of Algebra A Review 26 scienti c Notation The expression n 10k where n is a number greater than or equal to 1 and less than 10 written in decimal form and k is any integer is commonly called scienti c nota tion or the scienti c form of a number The following are examples of numbers ex pressed in scienti c form 4 23 104 8 176 1012 5 02 1023 1 1025 Very large and very small numbers can be conveniently expressed in scien ti c notation For example a light year the distance that a ray of light travels in one year is approximately 5 900 000 000 000 miles and this can be written as 5 9 1012 The weight of an oxygen molecule is approximately 0 000000000000000000000053 of a gram and this can be expressed as 5 3 10223 To change from ordinary decimal notation to scienti c notation the following procedure can be used Write the given number as the product of a number greater than or equal to 1 and less than 10 and a power of 10 The exponent of 10 is determined by counting the number of places that the decimal point was moved when going from the original number to the number greater than or equal to 1 and less than 10 This exponent is a
177. yB and simplify where possible solution 22xA 26x 1 218xyB 5 212x2 1 236x2y 5 24x223 1 236x22y 5 2x23 1 6x2y Rationalize radical expressions section 0 6 Objective 5 If a radical appears in the de nominator then it will be neces sary to rationalize the denomina tor for the expression to be in simplest form To rationalize a binomial denominator multiply the nu merator and denominator by the conjugate of the denominator The factors a 2 b and a 1 b are called conjugates Simplify 3 27 2 25 solution 3 27 2 25 5 3 A 27 2 25B A 27 1 25B A 27 1 25B 5 3A 27 1 25B 249 2 225 5 3A 27 1 25B 7 2 5 5 3A 27 1 25B 2 OBjecTiVe sUMMARy eXAMPLe Evaluate a number raised to a rational exponent section 0 7 Objective 1 If b is a real number n is a posi tive integer greater than 1 and 2 n b exists then b1yn 5 2 n b Thus b1yn means the nth root of b Simplify 163y2 solution 163y2 5 161y2 3 5 43 5 64 Simplify expressions with rational exponents section 0 7 Objective 2 Properties of exponents are used to simplify products and quo tients involving rational expo nents Simplify 4x1y3 23x3y4 and express the result with positive exponents only solution 4x1y3 23x23y4 5 212x1y323y4 5 212x25y12 5 212 x5y12 90360_ch0_001 102 indd 98 11 17 11 8 49 AM Copyright 2012 Cengage Learning All Rights Reserved May not be copied scanne

College Algebra (Kaufmann), 8th ed.

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