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Solution Manual for College Algebra, 4th Edition

Solution Manual for College Algebra, 4th Edition offers step-by-step solutions to help you understand tough concepts with ease.

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    I NSTRUCTOR ’ S S OLUTION M ANUAL B EVERLY F USFIELD C OLLEGE A LGEBRA F OURTH E DITION J. S. Ratti University of South Florida Marcus McWaters University of South Florida Leslaw A. Skrz y pek University of South Florida

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    CONTENTS Chapter P Basic Concepts of Algebra P.1 The Real Numbers and Their Properties ................................ ...................... 1 P.2 Integer Exponents and Scientific Notation ................................ ................... 7 P.3 Polynomials ................................ ................................ ................................ 13 P.4 Factoring Polynomials ................................ ................................ ................ 20 P.5 Rational Expressions ................................ ................................ .................. 27 P.6 Rational Exponents and Radicals ................................ ............................... 37 Chapter P Review Exercises ................................ ................................ .................... 46 Chapter P Practice Test ................................ ................................ ........................... 51 Chapter 1 Equations and Inequalities 1.1 Linear Equations in One Variable ................................ .............................. 53 1.2 Applications of Linear Equations: Modeling ................................ ............. 63 1. 3 Quadratic Equations ................................ ................................ ................... 7 4 1. 4 Complex Numbers : Quadratic Equations with Complex Solutions ........... 89 1 . 5 Solving Other Types of Equations ................................ ............................. 96 1 . 6 Inequalities ................................ ................................ ............................... 116 1 . 7 Equations and Inequalities Involving Absolute Value ............................. 1 3 1 Chapter 1 Review Exercises ................................ ................................ .................. 1 4 8 Chapter 1 Practice Test A ................................ ................................ ...................... 1 59 Chapter 1 Practice Test B ................................ ................................ ...................... 1 6 0 Chapter 2 Graphs and Functions 2.1 The Coordinate Plane ................................ ................................ ............... 163 2.2 Graphs of Equations ................................ ................................ ................. 172 2.3 Lines ................................ ................................ ................................ ......... 185 2.4 Functions ................................ ................................ ................................ .. 200 2.5 Properties of Functions ................................ ................................ ............. 210 2. 6 A Library of Functions ................................ ................................ ............. 219 2. 7 Transformations of Functions ................................ ................................ .. 230 2. 8 Combining Functions; Composite F unctions ................................ ........... 248 2. 9 Inverse Functions ................................ ................................ ..................... 26 3 Chapter 2 Review Exercises ................................ ................................ .................. 27 5 Chapter 2 Practice Test A ................................ ................................ ...................... 28 5 Chapter 2 Practice Test B ................................ ................................ ...................... 28 6 Cumulative Review Exercises (Chapters P−2) ................................ ...................... 28 7 Chapter 3 Polynomial and Rational Functions 3.1 Quadratic Functions ................................ ................................ ................. 29 1 3.2 Polynomial Functions ................................ ................................ ............... 310 3.3 Dividing Polynomials ................................ ................................ ............... 32 3 3.4 The Real Zeros of a Polynomial Function ................................ ................ 334 3.5 The Complex Zeros of a Polynomial Function ................................ ........ 353 3.6 Rational Functions ................................ ................................ .................... 363 3. 7 Variation ................................ ................................ ................................ ... 382 Chapter 3 Review Exercises ................................ ................................ .................. 387 Chapter 3 Practice Test A ................................ ................................ ...................... 403 Chapter 3 Practice Test B ................................ ................................ ...................... 404 Cumulative Review Exercises (Chapters P − 3) ................................ ...................... 405

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    Chapter 4 Exponential and Logarithmic Functions 4 .1 Exponential Functions ................................ ................................ .............. 408 4 . 2 Logarithmic Functions ................................ ................................ ............. 420 4 . 3 Rules of Logarithms ................................ ................................ ................. 433 4 . 4 Exponential and Logarithmic Equations and Inequalities ........................ 446 4.5 Logarithmic Scales ; Modeling ................................ ................................ . 459 Chapt er 4 Review Exercises ................................ ................................ .................. 470 Chapter 4 Practice Test A ................................ ................................ ...................... 47 9 Chapter 4 Practice Test B ................................ ................................ ...................... 4 80 Cumulative Review Exercises (Chapters P − 4 ) ................................ ...................... 480 Chapter 5 Systems of Equations and Inequalities 5 .1 Systems of Linear Equations in Two Variables ................................ ....... 483 5 .2 Systems of Li near Equations in Three Variables ................................ ..... 501 5 .3 Partial - Fraction Decomposition ................................ ............................... 522 5 .4 Systems of Nonlinear Equations ................................ .............................. 545 5 .5 Systems of Inequalities ................................ ................................ ............. 559 5 .6 Linear Programming ................................ ................................ ................. 572 Chapter 5 Review Exercises ................................ ................................ .................. 587 Chapter 5 Practice Test A ................................ ................................ ...................... 600 Chapter 5 Practice Test B ................................ ................................ ...................... 603 Cumulative Review Exercises (Chapters P − 5 ) ................................ ...................... 605 Chapter 6 Matrices and Determinants 881 6 .1 Matrices and Systems of Equations ................................ .......................... 608 6 .2 Matrix Algebra ................................ ................................ ......................... 6 2 9 6 .3 The Matrix Inverse ................................ ................................ ................... 649 6 .4 Determinants and Cramer’s Rule ................................ ............................. 670 Chapter 6 Review Exercises ................................ ................................ .................. 683 Chapter 6 Practice Test A ................................ ................................ ...................... 694 Chapter 6 Practice Test B ................................ ................................ ...................... 695 Cumulative Review Exercises (Chapters P − 6 ) ................................ ...................... 696 Chapter 7 The Conic Sections 7 .2 The Parabola ................................ ................................ ............................. 699 7 .3 The Ellipse ................................ ................................ ................................ 714 7 .4 The Hyperbola ................................ ................................ .......................... 732 Chapter 7 Review Exercises ................................ ................................ .................. 756 Chapter 7 Practice Test A ................................ ................................ ...................... 765 Chapter 7 Practice Test B ................................ ................................ ...................... 767 Cumulative Review Exercises (Chapters P − 7 ) ................................ ...................... 768 Chapter 8 Further Topics in Algebra 8 .1 Sequences and Series ................................ ................................ ................ 771 8 .2 Arithmetic Sequences; Partial Sums ................................ ........................ 782 8 .3 Geometric Sequences and Series ................................ .............................. 790 8 .4 Mathematical Induction ................................ ................................ ............ 799 8 .5 The Binomial Theorem ................................ ................................ ............ 810 8 .6 Counting Principles ................................ ................................ .................. 818 8 .7 Probability ................................ ................................ ................................ 825 Chapter 8 Review Exercises ................................ ................................ .................. 829 Chapter 8 Practice Test A ................................ ................................ ...................... 833 Chapter 8 Practice Test B ................................ ................................ ...................... 833 Cumulative Review Exercises (Chapters P − 8 ) ................................ ...................... 834

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    1 Chapter P Basic Concepts of Algebra P .1 The Real Numbers and Their Properties P.1 Practice Problems 1. Let x = 2.132132132…. Then, 1000 x = 2132.132132… 1000 2132.132132 2.132132 999 2130 2130 710 999 333 x x x x = = = = = K K 2. a. Natural numbers: 2, 7 b. Whole numbers: 0, 2, 7 c. Integers: 21 6, 3, 0, 2, 7 7 - - = - d. Rational numbers: 21 1 4 6, 3, , 0, , 2, 7 7 2 3 - - = - - e. Irrational numbers: 3, 17 f. Real numbers: the set B 3 . a. 3 2 8 = b. ( ) ( )( ) 2 2 3 3 3 9 = = a a a a c. 4 1 1 1 1 1 1 2 2 2 2 2 16 æ ö = × × × = ç ÷ è ø 4 . a. True b. True c. False 5 . { } { } 3, 1, 0, 1, 3 , 4, 2, 0, 2, 4 = - - = - - A B { } { } 0 , 4, 3, 2, 1, 0, 1, 2, 3, 4 = = - - - - A B A B I U 6 . a. ( ) 1 2 , I I = - ¥ ¥ U b. [ ) 1 2 2, 5 I I = - I 7 . a. 10 10 - = b. 3 4 1 1 - = - = c. ( ) 2 3 7 1 1 - + = = 8 . ( ) 7, 2 7 2 9 9 - = - - = - = d 9 . a. ( ) 3 5 20 15 20 5 - × + = - + = b. 5 12 6 2 5 2 2 5 4 1 - ¸ × = - × = - = c. 9 1 8 5 7 5 7 2 35 33 4 4 - - × = - × = - = - d. ( ) 2 3 4 x - + - for x = 6 ( ) 2 2 3 6 4 3 2 3 4 1 - + - = - + = - + = 10 . a. 7 3 14 3 17 4 8 8 8 8 + = + = b. 8 2 40 6 34 3 5 15 15 15 - = - = c. 9 7 9 14 3 × = 3 14 2 7 × 1 3 1 3 2 = d. 5 5 15 5 16 5 8 15 8 16 8 15 16 = ¸ = × = 1 8 1 16 × 2 15 3 2 3 = 11 . a. ( ) ( ) 2 3 3 2 3 3 1 10 1 3 3 3 3 3 x x - ¸ + = - ¸ + = ¸ + = + = b. 1 1 22 7 7 7 3 3 3 x y - - = - = + = 1 2 . 39 C 4 13 4 17 C 3 ° = + = + = ° P .1 Concepts and Vocabulary 1. Whole numbers are formed by adding the number zero to the set of natural numbers. 2. The number −3 is an integer, but it is also a rational number and a real number . 3. If a < b , then a is to the left of b on the number line. 4. If a real number is not a rational number, it is an irrational number.

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    2 Chapter P Basic Concepts of Algebra 5. True 6. False. 5 1 2 2 2 - = - 7. True 8. False. An example is ( ) ( ) 2 2 2 2 4, + + - = which is rational. P.1 Building Skills 9 . 0.3 , repeating 10 . 0.6 , repeating 11 . 0.8 - , terminating 1 2 . 0.25 - , terminating 1 3 . 0.27 , repeating 1 4 . 0.3 , repeating 1 5 . 3.16 , repeating 1 6 . 2.73 , repeating 1 7 . 375 15 3.75 100 4 = = 1 8 . 235 47 2.35 100 20 - = - = - 1 9 . 10 53.3 5.3 9 48 48 16 9 3 16 5.3 3 x x x x = = = = = - = - 20 . 10 96.6 9.6 9 87 87 29 9 3 x x x x = = = = = 21 . 100 213.13 2.13 99 211 211 99 x x x x = = = = 22 . 100 323.23 3.23 99 320 320 99 x x x x = = = = 23 . 100 452.323 4.523 99 447.8 447.8 4478 2239 99 990 495 x x x x = = = = = = 24 . 100 142.3535 1.4235 99 140.93 140.93 14, 093 99 9900 x x x x = = = = = 25 . R ational 26 . R ational 27 . R ational 28 . R ational 29 . R ational 30 . R ational 31 . I rrational 32 . I rrational Exercises 33 - 38 refer to the set 12 17 A 19, , 3, 0, 2, 10, , 11 3 4 ì ü = - - - í ý î þ 33. Natural numbers: 2, 11 34. Whole numbers: 0, 2, 11 35. Integers: 12 19, 4, 0, 2, 11 3 - - = - 36. Rational numbers: 12 17 19, , 0, 2, , 11 3 4 - - 37. Irrational numbers: 3, 10 - 38. Real numbers: All numbers in set A are real numbers. 39. 3 10 base: 10; exponent: 3 3 10 10 10 10 1000 = × × = 40. 4 5 base: 5; exponent: 4 4 5 5 5 5 5 625 = × × × = 41. 3 2 3 æ ö ç ÷ è ø base: 2 ; 3 exponent: 3 3 2 2 2 2 8 3 3 3 3 27 æ ö = × × = ç ÷ è ø 42. 4 5 2 æ ö ç ÷ è ø base: 5 ; 2 exponent: 4 4 5 5 5 5 5 625 2 2 2 2 2 16 æ ö = × × × = ç ÷ è ø

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    Section P. 1 The Real Numbers and Their Properties 3 43. ( ) 3 2 - base: – 2; exponent: 3 ( ) ( ) ( ) ( ) 3 2 2 2 2 8 - = - × - × - = - 44. 4 1 2 æ ö - ç ÷ è ø base: 1 ; 2 - exponent: 4 4 1 1 1 1 1 1 2 2 2 2 2 16 æ ö æ öæ öæ öæ ö - = - - - - = ç ÷ ç ÷ç ÷ç ÷ç ÷ è ø è øè øè øè ø 45. 4 2 3 × base: 3; exponent: 4 4 2 3 2 3 3 3 3 2 81 162 × = × × × × = × = 46. 3 1 5 3 æ ö ×ç ÷ è ø base: 1 ; 3 exponent: 3 3 1 1 1 1 1 5 5 5 5 3 3 3 3 27 27 æ ö × = × × × = × = ç ÷ è ø 47. 4 2 3 - × base: 3; exponent: 4 4 2 3 2 3 3 3 3 2 81 162 - × = - × × × × = - × = - 48. ( ) 4 3 2 - × - base: – 2; exponent: 4 ( ) ( )( )( )( ) 4 3 2 3 2 2 2 2 3 16 48 - × - = - - - - - = - × = - 49. ( ) 5 3 2 - × - base: – 2; exponent: 5 ( ) ( )( )( )( )( ) ( ) 5 3 2 3 2 2 2 2 2 3 32 96 - × - = - - - - - - = - × - = 50. ( ) 2 5 3 - × - base: – 3; exponent: 2 ( ) ( )( ) 2 5 3 5 3 3 5 9 45 - × - = - - - = - × = - 51 . 3 2 > - 52 . 3 2 - < - 53 . 1 1 2 2 ³ 54 . 1 < + x x 55 . 5 2 £ x 56 . 1 2 - > x 5 7 . 0 x - > 5 8 . 0 x < 5 9 . 2 7 14 + £ x 6 0 . 2 3 5 + £ x 6 1 . 24 4 6 = 6 2 . 5 2 - < - 6 3 . 4 0 - < 6 4 . 9 1 4 2 2 - = - 6 5. { } 4, 3, 2, 0, 1, 2, 3, 4 = - - - A B U 6 6. { } 0, 2, 4 = A B I 6 7 . { } 4, 2, 0, 2 A C = - - I 6 8 . { } 4, 3, 2, 1, 0, 1, 2, 3, 4 B C = - - - - U 6 9 . ( ) { } { } 3, 0, 2 4, 3, 2, 0, 2, 4 B C A A = - = - - - I U U 7 0 . ( ) { } { } 4, 3, 2, 1, 0, 2, 4 3, 0, 2, 4 A C B B = - - - - = - U I I 7 1 . ( ) { } { } 4, 3, 2, 0, 1, 2, 3, 4 4, 3, 2, 0, 2 A B C C = - - - = - - - U I I 7 2 . ( ) { } { } 4, 3, 2, 0, 1, 2, 3, 4 4, 3, 2, 1, 0, 1, 2, 3, 4 A B C C = - - - = - - - - U U U 7 3. ( ) [ ] 1 2 1 2 2, 5 ; 1, 3 I I I I = - = U I 7 4. [ ] ( ) 1 2 1 2 1, 7 ; 3, 5 I I I I = = U I 7 5. ( ) 1 2 1 2 6, 10 ; I I I I = - = Æ U I 7 6. ( ) 1 2 1 2 , ; I I I I = - ¥ ¥ = Æ U I 7 7. ( ) [ ) 1 2 1 2 , ; 2, 5 I I I I = - ¥ ¥ = U I 7 8. ( ) ( ) 1 2 1 2 2, ; 0, I I I I = - ¥ = ¥ U I 7 9. ( ) [ ) [ ] 1 2 1 2 , ; 1, 3 5, 7 I I I I = - ¥ ¥ = - U I U 8 0. ( ] ( ) [ ] 1 2 1 2 , 2 6, ; 3, 0 I I I I = - ¥ ¥ = - U U I 8 1 . 4 4 - = 8 2 . 17 17 - - = - 8 3 . 5 5 7 7 = - 8 4 . 3 3 5 5 - = 8 5 . 5 2 5 2 - = - 8 6 . 2 5 5 2 - = - 8 7. 3 2 2 3 - = - 8 8. 3 3 p p - = - 8 9 . 8 8 1 8 8 = = - 9 0 . 8 8 1 8 8 - - = = -

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    4 Chapter P Basic Concepts of Algebra 9 1 . 5 7 5 7 2 2 - - = - = - = 9 2 . 4 7 4 7 3 3 - = - = - = 9 3 . (3,8) 3 8 5 5 = - = - = d 9 4 . (2,14) 2 14 12 12 = - = - = d 9 5 . ( 6,9) 6 9 15 15 - = - - = - = d 9 6 . ( 12,3) 12 3 15 15 - = - - = - = d 9 7 . ( 20, 6) 20 ( 6) 14 14 - - = - - - = - = d 9 8 . ( 14, 1) 14 ( 1) 13 13 - - = - - - = - = d 9 9 . 22 4 22 4 26 26 , 7 7 7 7 7 7 æ ö æ ö - = - - = = ç ÷ ç ÷ è ø è ø d 10 0 . 16 3 16 3 19 19 , 5 5 5 5 5 5 æ ö æ ö - = - - = = ç ÷ ç ÷ è ø è ø d 10 1 . 3 1 - < £ x 10 2 . 6 2 - £ < - x 10 3 . 3 ³ - x 10 4 . 0 ³ x 10 5 . 5 £ x 10 6 . 1 £ - x 10 7 . 3 9 4 4 - < < x 10 8 . 1 3 2 - < < - x 10 9. 4( 1) 4 4 + = + x x 11 0. ( 3)(2 ) 6 3 - - = - + x x 11 1 . 5( 1) 5 5 5 - + = - + x y x y 11 2 . 2(3 5 ) 6 10 2 + - = + - x y x y 11 3. A dditive inverse: 5 - ; reciprocal: 1 5 11 4 . A dditive inverse: 2 3 ; reciprocal: 3 2 - 11 5 . A dditive inverse: 0 ; no reciprocal 11 6. A dditive inverse: 1.7 - ; reciprocal: 10 17 11 7. A dditive inverse 11 8. A dditive inverse 11 9. M ultiplicative identity 1 2 0. M ultiplicative identity 1 2 1. A ssociative property of multiplication

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    Section P. 1 The Real Numbers and Their Properties 5 1 2 2. A ssociative property of multiplication 1 2 3. M ultiplicative inverse 1 2 4 . M ultiplicative inverse 1 2 5. A dditive identity 1 2 6. A dditive identity 1 2 7. A ssociative property of addition 1 2 8. C ommutative and associative propert ies of addition 1 2 9. 3 4 9 20 29 5 3 15 15 15 + = + = 1 3 0. 7 3 14 15 29 10 4 20 20 20 + = + = 1 3 1. 6 5 42 25 67 5 7 35 35 35 + = + = 1 3 2. 9 5 54 5 59 2 12 12 12 12 + = + = 1 3 3. 5 3 25 9 34 17 6 10 30 30 30 15 + = + = = 1 3 4. 8 2 24 10 34 15 9 45 45 45 + = + = 1 3 5. 5 9 25 36 11 8 10 40 40 40 - = - = - 1 3 6. 7 1 35 8 27 8 5 40 40 40 - = - = 1 3 7. 5 7 55 63 8 9 11 99 99 99 - = - = - 1 3 8. 5 7 15 14 1 8 12 24 24 24 - = - = 1 3 9. 2 1 4 5 1 5 2 10 10 10 - = - = - 1 4 0. 1 1 3 2 1 4 6 12 12 12 - = - = 1 4 1. 3 8 2 4 27 9 × = 1 4 2. 9 14 2 7 27 3 × = 1 4 3. 8 8 16 8 15 3 5 16 5 15 5 16 2 15 = ¸ = × = 1 4 4. 5 5 15 5 6 1 6 15 6 6 6 15 3 6 = ¸ = × = 1 4 5. 7 7 21 7 16 2 8 21 8 16 8 21 3 16 = ¸ = × = 1 4 6. 3 3 7 3 15 9 10 7 10 15 10 7 14 15 = ¸ = × = 1 4 7. 3 1 3 1 2 5 1 10 2 2 2 2 × - = - = = 1 4 8. 7 3 14 3 11 2 2 2 2 2 2 × - = - = 1 4 9. 2 1 2 1 6 5 1 3 15 3 5 3 15 15 15 × - = - = - = 1 5 0. 5 3 10 3 20 9 11 2 3 2 3 2 6 6 6 × - = - = - = 1 5 1. 2( ) 3 2(3 ( 5)) 3( 5) 2( 2) ( 15) 4 15 11 + - = + - - - = - - - = - + = x y y 1 5 2. 2( ) 5 2(3 ( 5)) 5( 5) 2( 2) ( 25) 4 ( 25) 21 - + + = - + - + - = - - + - = + - = - x y y 1 5 3. 3 2 3 3 2 5 3(3) 2(5) 9 10 1 - = - - = - = - = - x y 1 5 4. 7 7 3 ( 5) 7 8 7(8) 56 - = - - = = = x y 1 5 5. 3 3 3( 5) 3( 5) 2 2 3 ( 15) ( 15) 2 18 ( 15) 9 ( 15) 6 2 - - - + = + - - - = + - = + - = + - = - x y xy 1 5 6. 3 5 3 3( 5) 3 2 43 ( 15) 3 3 + - + - = - - - = - - = y xy x 1 5 7. 2(1 2 ) 2(1 2(3)) ( ) ( 3)( 5) 5 2( 5) 15 2 15 13 5 x x y y - - - - = - - - - - = - = - = - -

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    6 Chapter P Basic Concepts of Algebra 1 5 8 . 3(2 ) 3(2 3) (1 ) (1 3( 5)) 5 3( 1) (1 ( 15)) 5 3 77 16 5 5 - - - - = - - - - - = - - - - = - = - x xy y 1 5 9 . 14 1 14 1 31 4 62 2 3 2 ( 5) 6 5 15 4 4 + + = = × = - - - x y 1 6 0 . 4 8 4 8 4 8 2 ( 5) 3 5 5 3 5 2 2 52 2 104 15 5 75 + + - - - æ ö æ ö = = + × - ç ÷ ç ÷ - è ø è ø æ ö = × - = - ç ÷ è ø y x y 1 6 1 . 3 3 3 x x x xy y y + + = 1 6 2 . ( )( ) 2 2 2 3 3 2 6 5 6 x x x x x x x + + = + + + = + + 1 6 3 . ( ) ( ) 5 3 5 5 3 5 15 x x x + = + = + . 1 6 4. ( )( ) 2 25 4 100 x x x = 1 6 5. ( ) 3 2 3 2 x y x y - + = - - 1 6 6. ( ) 2 4 5 2 4 5 x y x y - - = - + 1 6 7. 1 x y x y y x x x x + = + = + 1 6 8. x y x y x z x z x z + = + + + + 1 6 9. ( )( ) 1 1 1 x y xy x y + + = + + + 1 7 0. 1 1 1 1 1 1 y y x x y x x y = ¸ = × = 1 7 1. ( ) ( ) 2 , 2 2 2 2 , 2 2 2 a b a b a a b d P M a a b b a b b a d Q M b + + - - + = - = = + - + - = - = = Since ( ) ( ) , , , 2 2 a b b a d P M d Q M - + - = = = M is the midpoint of the line segment PQ . 1 7 2. Answers may vary. Using the hint, we have 4 40 13 130 - = - and 7 70 . 13 130 = Therefore, 40 31 30 0 130 130 130 1 10 69 70 . 130 130 130 130 - < - < - < < < < < < < < P.1 Applying the Concepts 1 7 3 . a. people who own either MP3 players or people who own DVD players. b. people who own both MP3 players and DVD players. 1 7 4. a. A = {Lexus LS 4 6 0} b. B = {Lexus LS 46 0, Lincoln Continental } c. C = {Lexus LS 460 , Lincoln Continental , Audi A 4 } d. A B I = {Lexus LS 460 } e. B C I = {Lexus LS 460 , Lincoln Continental } f. A B U = {Lexus LS 460 , Lincoln Continental } g. A C U = {Lexus LS 460 , Lincoln Continental , Audi A 4 } 1 7 5. 119.5 134.5 £ £ x 1 7 6. 30 107 £ £ x 1 7 7. a. 124 120 4 - = b. 137 120 17 - = c. 114 120 6 6 - = - = 1 7 8 . a. 15 14 1 - = b. 15 17.5 2.5 - = c. 15 9 6 - = 1 7 9 . Let x = the number of calories from broccoli. Then we have 522.5 55 0 522.5 55 9.5 - = Þ = Þ = x x x The number of grams of broccoli is 9.5 × 100 = 950 grams.

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    Section P. 2 Integer Exponents and Scientific Notation 7 1 8 0 . Let x = the number of orders of f rench fries. The number of calories lost from broccoli is 6 × 55 = 330. Then we have 165 330 0 165 330 2 - = Þ = Þ = x x x So, Carmen will have to eat 2 orders of french fries. P.1 Beyond the Basics 181. True 182. True 183. a. False. For example, 0 2 0. × = b. The products of a nonzero rational number and an irrational number is an irrational number. 184. False. For example , 2 2 2. × = 185. False. For example , ( ) ( ) 2 3 6 3 4. + - + = - 186. False. For example, 12 4 2. 3 = = 187. True 188. True For exercises 189 – 190, use the following definition: An integer P is even if p = 2 n for some integer n . An integer q is odd if q = 2 k + 1 for some integer k . 189. a. If a is odd, then a = 2 m + 1 for some integer m . ( ) ( )( ) ( ) ( ) ( ) 2 2 2 2 1 2 1 2 1 4 4 1 4 1 1 2 2 1 1, a m m m m m m m m m = + = + + = + + = + + = + + which is of the format 2 k +1. Therefore, 2 a is an odd integer. b. Assume that b is odd. Then, from part (a), 2 b is also odd. However, 2 b is an even integer , so, by contradiction, b must be an even integer. 190. 2 2 q is even because it is of the form 2 n , so it follows that 2 p is even. Then, from exercise 189(b), p is also even. Therefore, p = 2 n , for some integer n . Substituting, we have ( ) 2 2 2 2 2 2 2 2 2 2 q p n q n = = Þ = × Þ 2 2 2 , q n = and, thus, q is even using exercise 189 (b). P.1 Getting Ready for the Next Section 191. a. 5 2 3 a a a × = b. 4 7 11 a a a × = c. m n m n a a a + × = 192. a. 3 2 b b b = b. 7 4 3 b b b = c. m m n n b b b - = 193. a. ( ) 3 6 2 a a = b. ( ) 2 8 4 a a = c. ( ) n m mn a a = 194. a. ( ) 2 2 2 ab a b = b. ( ) 4 4 4 ab a b = c. ( ) n n n ab a b = P.2 Integer Exponents and Scientific Notation P.2 Practice Exercises 1 . a. 1 1 2 2 - = b. 0 4 1 5 æ ö = ç ÷ è ø c. 2 2 3 2 4 2 3 9 - æ ö æ ö = = ç ÷ ç ÷ è ø è ø 2 . a. 2 7 9 3 3 × = x x x b. ( )( ) ( ) 3 3 2 3 3 3 3 0 2 4 4 4 16 16 16 1 16 x x x x x x + - - - = × = = = × = 3 . a. 4 4 0 4 0 3 3 3 81 3 - = = =

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    8 Chapter P Basic Concepts of Algebra b. 1 ( 2) 3 2 5 5 5 125 5 - - - = = = c. 3 3 ( 4) 7 4 2 2 2 3 3 3 - - - = = x x x x 4 . a. ( ) 0 5 ( 5)(0) 0 7 7 7 1 - - = = = b. ( ) 5 0 (0)( 5) 0 7 7 7 1 - - = = = c. ( ) 8 1 ( 1)(8) 8 8 1 x x x x - - - = = = d. ( ) 5 2 ( 2)( 5) 10 - - - - = = x x x 5 . a. ( ) 1 1 1 1 1 1 2 2 2 2 - - - æ ö æ ö æ ö = = = ç ÷ ç ÷ ç ÷ è ø è ø è ø x x x x b. ( ) ( ) 2 2 1 2 1 2 ( 1)(2) 2 2 5 5 5 25 25 - - - - = = = = x x x x x c. ( ) 3 2 3 2(3) 3 6 = = xy x y x y d. ( ) 6 3 2 ( 2)( 3) 3 3 - - - - - = = x x y x y y 6 . a. 2 2 2 1 1 1 3 9 3 æ ö = = ç ÷ è ø b. 2 2 2 2 10 7 7 49 7 10 100 10 - æ ö æ ö = = = ç ÷ ç ÷ è ø è ø 7 . a. ( ) 2 2 4 4 8 1 1 2 2 4 - æ ö = = ç ÷ è ø x x x b. ( ) ( ) 3 2 2 3 3 3 6 3 2 1 - - = = - x y x y x y xy xy 8 . 5 732,000 7.32 10 = ´ 9 . 10 2 2 8 2.9 10 290 10 0.89 10 $89 325 3.25 10 ´ = ´ » ´ » ´ per person P . 2 Concepts and Vocabulary 1. In the expression 2 7 , the number 2 is called the exponent . 2. In the expression 7 3 , - the base is 3 . 3. The number 2 1 4 - simplifies to be the positive integer 16 . 4. The power - of - a - product rule allows us to rewrite ( ) 3 5 a as 3 3 5 . a 5. False. ( ) 10 10 11 11 - = 6. False. When ( ) 3 2 x is simplified, the expression becomes 6 . x 7. True 8. False P.2 Building Skills 9. 2 2 1 1 1 1 3 3 3 3 9 - æ ö æ öæ ö = = = ç ÷ ç ÷ç ÷ è ø è øè ø 10. 3 3 1 1 1 1 1 2 2 2 2 2 8 - æ ö æ öæ öæ ö = = = ç ÷ ç ÷ç ÷ç ÷ è ø è øè øè ø 11. 4 4 1 2 16 2 - æ ö = = ç ÷ è ø 12. ( ) 2 2 1 2 4 2 - æ ö - = - = ç ÷ è ø 13. 0 7 1 = 14. ( ) 0 8 1 - = 15. ( ) 0 7 1 - - = - 16. ( ) 0 2 1 = 17 . ( ) 2 3 3 2 6 2 2 2 64 × = = = 18 . ( ) 3 2 2 3 6 3 3 3 729 × = = = 19 . ( ) ( ) 2 2 2 2 4 4 1 1 3 3 3 81 3 - - - = = = = 20 . ( ) ( ) 1 2 1 2 2 2 1 1 7 7 7 49 7 - - - = = = =

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    Section P. 2 Integer Exponents and Scientific Notation 9 21 . ( ) 3 3 3 2 2 1 1 1 5 25 15, 625 5 - æ ö æ ö = = = ç ÷ ç ÷ è ø è ø 22 . ( ) 3 3 3 1 1 1 1 5 5 5 125 - æ ö æ ö = = = ç ÷ ç ÷ è ø è ø 23 . ( ) ( ) 3 5 ( 3 5) 2 4 4 4 4 16 - - + × = = = 24 . ( ) ( ) 2 3 ( 2 3) 1 7 7 7 7 7 - - + × = = = 25 . ( ) 0 0 3 10 1 1 2 + = + = 26 . ( ) 0 0 5 9 1 1 0 - = - = 27 . 2 2 2 2 1 1 1 1 1 2 3 3 3 3 9 9 9 - æ ö æ ö æ ö + = + = + = ç ÷ ç ÷ ç ÷ è ø è ø è ø 28 . 2 2 2 2 1 1 1 1 1 2 5 5 5 5 25 25 25 - æ ö æ ö æ ö + = + = + = ç ÷ ç ÷ ç ÷ è ø è ø è ø 29 . 3 3 1 1 2 8 2 - - = - = - 30 . 2 2 1 1 3 9 3 - - = - = - 31 . ( ) ( ) 2 2 1 1 3 9 3 - - = = - 32 . ( ) ( ) 3 3 1 1 2 8 2 - - = = - - 33 . 11 (11 10) 1 10 2 2 2 2 2 - = = = 34 . 6 (6 8) 2 8 2 3 1 1 3 3 9 3 3 - - = = = = 35 . ( ) 4 3 12 12 12 5 5 1 5 5 = = 36 . ( ) 2 5 10 (10 8) 2 8 8 9 9 9 9 81 9 9 - = = = = 37 . 5 2 (5 4) ( 2 ( 3)) 1 1 4 3 2 3 2 3 2 3 6 2 3 - - - - - - × = × = × = × 38 . 2 3 ( 2 ( 3)) (3 1) 3 1 2 4 5 4 5 4 5 4 5 4 25 100 - - - - - - × = × × = × = × = 39 . 2 1 1 2 5 1 1 2 2 2 25 25 2 5 - - - = - × = - × = - 40 . 2 1 1 2 7 1 1 3 3 3 49 49 3 7 - - - = - × = - × = - 41. 1 2 3 3 2 - æ ö = ç ÷ è ø 42. 1 1 5 5 - æ ö = ç ÷ è ø 43. 2 2 2 2 2 3 3 9 3 2 4 2 - æ ö æ ö = = = ç ÷ ç ÷ è ø è ø 44. 2 2 2 2 3 2 2 4 2 3 9 3 - æ ö æ ö = = = ç ÷ ç ÷ è ø è ø 45 . 2 2 11 1 1 49 121 7 121 11 49 7 - æ ö = = = ç ÷ è ø æ ö ç ÷ è ø 46 . 2 2 13 1 1 25 169 5 169 13 25 5 - æ ö = = = ç ÷ è ø æ ö ç ÷ è ø 47 . 4 0 4 4 1 = × = x y x x 48 . 1 0 1 1 1 - = × = x y x x 49 . 1 1 - = × = y x y y x x 50 . 2 2 2 2 2 2 1 - = × = x x y x y y 51. 1 2 2 1 - - = x y xy 52. 3 2 3 2 1 - - = x y x y 53. ( ) 4 3 12 12 1 - - = = x x x 54. ( ) 2 5 10 10 1 - - = = x x x 55. ( ) 3 11 ( 11) ( 3) 33 x x x - - - ×- = = 56. ( ) 12 4 ( 4) ( 12) 48 x x x - - - ×- = =

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    10 Chapter P Basic Concepts of Algebra 57. 5 5 5 3( ) 3 - = - xy x y 58. 6 6 6 8( ) 8 - = - xy x y 59. ( ) 2 2 1 2 2 2 4 4 4 x xy x y y - - = = 60. ( ) 3 3 1 3 3 3 6 6 6 y x y x y x - - = = 61. ( ) 5 5 1 ( 1) ( 5) 5 5 3 3 3 x x y x y y - - - ×- - = = 62. ( ) 6 6 1 6 ( 1) ( 6) 6 5 5 5 y xy x y x - - - - × - - = - = - 63. ( ) ( ) 2 3 3 2 6 6 10 4 5 2 5 10 4 2 1 x x x x x x x x x × - - × = = = = = 64. ( ) 2 2 2 2 12 10 4 3 4 12 10 3 1 x x x x x x x x x - - × = = = = = 65. 3 3 3 3 3 3 3 6 3 2 2 3 6 3 3 3 3 2 2 8 8 8 8 - × - æ ö = = = ç ÷ è ø = = xy x y x y x y x x x y x y x 66. 4 4 4 4 4 4 4 12 4 3 3 4 12 4 8 4 8 5 5 625 625 625 625 - × - æ ö = = = ç ÷ è ø = = xy x y x y x y x x x y x y x 67. 5 2 5 2 5 5 10 5 5 5 10 5 5 5 5 3 ( 3) 243 243 243 × - æ ö - - - = = ç ÷ è ø = - = - x y x y x y x x x x y x y 68. 3 2 3 3 2 3 3 6 3 3 3 6 3 3 3 2 ( 2) 8 8 8 × - æ ö - - - = = ç ÷ è ø = - = - xy x y x y y y y x y x y 69. 2 2 2 2 2 2 2 1 1 1 3 ( 3) 25 9 9 1 1 5 5 9 25 25 x x x x x - - - - × - - æ ö = = = = ç ÷ è ø 70. 4 4 4 4 4 4 4 4 4 1 1 1 5 ( 5) 5 625 1 1 3 3 81 3 81 625 - - - - × - - æ ö = = = ç ÷ è ø = y y y y y 71. 3 2 3 2 3 6 5 3 5 3 3 15 9 15 4 4 64 64 x x x xy x y x y x y - - × - × æ ö = = = ç ÷ è ø 72. 5 2 5 2 5 5 10 5 10 3 3 5 15 10 3 3 243 243 × × æ ö = = = ç ÷ è ø x y x y x y x y y y y 73. 3 3 5 3 ( 2) 3 1 5 4 2 4 - - - - - - - = = = x y x x y x y x y y 74. 2 2 3 2 ( 1) 2 2 3 4 1 2 4 - - - - - - - = = = x y x x y x y x y y 75. 3 5 3 ( 4) 5 7 1 2 4 7 2 27 3 3 3 9 - - - - - - - = = = x y x x y x y x y y 76. 5 2 5 7 2 ( 3) 2 1 7 3 2 15 5 5 5 3 - - - - - - - = = = x y y x y x y x y x 77. ( ) ( ) ( ) 3 2 4 3 2 3 4 3 6 4 3 3 6 4 1 1 1 x x x x x x x x x x x x - - × - - - - + - - = = = = = 78. ( ) ( ) ( ) ( ) 12 12 12 4 3 4 3 4 4 12 4 3 12 4 12 12 12 3 4 12 4 12 12 12 12 12 12 12 4 12 12 12 12 12 4 12 0 0 16 16 16 2 2 8 8 2 2 2 2 2 2 2 1 2 a a a b a b b b a b a b a b a b a a b b a b a b b b - - - - - - - - - - - - - - - - - + - + - - - - æ ö = × ç ÷ è ø = × = × = × × × × × = × × = = =

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    Section P. 2 Integer Exponents and Scientific Notation 11 79. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 3 2 3 4 2 2 2 4 2 2 3 3 2 4 4 4 2 2 3 2 3 3 6 2 4 2 4 8 8 3 8 6 8 5 2 5 2 2 2 2 16 16 1 16 1 1 16 16 xy xy x y x y x y x y x y x y x y x y x y x y x y - × × × - - - - - - - = - - = - - × - = = = - = - = - 80. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 3 2 4 2 3 3 4 5 5 5 2 6 1 5 5 2 6 5 1 5 2 2 2 6 6 12 6 2 2 2 12 2 x y y x y y x y xy x y x y x y xy x y y x y x y x - - - × - - - - - - - - - - - - - - - - é ù é ù - - ê ú ê ú = ê ú ê ú ê ú ë û ë û æ ö - = ç ÷ è ø = - = - = - = = = 81. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 2 7 3 2 3 2 7 3 2 2 2 2 3 2 2 7 7 3 2 7 4 4 4 x y z x y z x y z x y z x y z x y z - = = 3 2 2 2 2 6 4 2 3 7 2 7 7 21 14 7 6 21 4 14 2 7 15 10 5 15 10 5 16 16 16 16 16 x y z x y z x y z x y z x y z x y z x y z × × × × - - - - - - = = = = = 82. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 3 2 2 1 1 6 2 1 1 1 3 2 2 2 2 1 2 5 2 2 5 2 1 3 3 0 3 3 2 2 4 4 4 4 4 xyz x y z x y z x y x z x y x z x y xz x y z x y z x y z z x y z x × - - - - - - - - - - - = = = = = = 83 . 2 2 2 4 1 ( 3) 2 2 4 3 2 4 6 4 6 5 5 5 5 1 - - - - - - - - = = × = a bc a b c a b c b a b a 84 . 2 5 2 5 2 2 5 2 2 3 2 4 3 2 3 4 2 3 4 3 3 1 2 2 ( 3) ( ) 9 9 8 ( 2) 8 9 9 8 8 - - - - - - = = - - - = - = - a bc a b c a b c a b c a b c a a b c bc 85 . 3 3 2 3 ( 3)( 3) ( 2)( 3) 2 4 3 2( 3) ( 4)( 3) 3( 3) 3 9 6 6 12 9 3 ( 6) 9 12 6 ( 9) 3 15 3 3 15 3 - - - - - - - - - - - - - - - - - - - - - - - æ ö = ç ÷ è ø = = = = xy z x y z x y z x y z x y z x y z x y z x z x y z y 86 . 1 2 1 1 ( 2)( 1) ( 1)( 1) 5 8 ( 5)( 1) 1 ( 8)( 1) 1 2 1 1 5 2 ( 1) 1 8 5 1 8 3 6 3 7 6 7 xy z x y z x yz x y z x y z x y z x y z y x y z x z - - - - - - - - - - - - - - - - - - - - - - - - æ ö = ç ÷ è ø = = = = 87 . 2 125 1.25 10 = ´ 88 . 2 247 2.47 10 = ´ 89 . 5 850,000 8.5 10 = ´ 90 . 5 205,000 2.05 10 = ´ 91 . 3 0.007 7.0 10 - = ´ 92 . 3 0.0019 1.9 10 - = ´ 93. 6 0.00000275 2.75 10 - = ´ 94 . 6 0.0000038 3.8 10 - = ´ P.2 Applying the Concepts 95 . 3 3 3 135 ft 2 1080 ft ´ = 96 . 3 3 3 1 675 in. 25 in. 3 æ ö ´ = ç ÷ è ø 97 . a. 2 2 2 (2 ) 4 4 2 = = = x x A A b. 2 2 2 (3 ) 9 9 3 = = = x x A A

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    12 Chapter P Basic Concepts of Algebra 98 . a. 2 2 2 2 2 4 2 2 2 p p p æ ö æ ö = = = ç ÷ ç ÷ è ø è ø d d d A b. 2 2 2 2 3 9 9 3 2 2 2 p p p æ ö æ ö æ ö = = = ç ÷ ç ÷ ç ÷ è ø è ø è ø d d d A 99 . 2 2 25, 000(0.25) 1562.5 lb = = = F Sw 100 . 2 2 1.5 10, 000(3.14) 2 2 17, 662.5 lbs. p æ ö æ ö = = ç ÷ ç ÷ è ø è ø » d F S 101. 7 24 hr 60 min 60 sec 365.25 days day hr min 365.25 24 60 60 sec 31,557,600 sec 3.15576 10 sec ´ ´ ´ = ´ ´ ´ = = ´ 102. 7 24 hr 60 min 60 sec 366 days day hr min 366 24 60 60 sec 31,622,400 sec 3.16224 10 sec ´ ´ ´ = ´ ´ ´ = = ´ 10 3 . Celestial Body Equatorial Diameter (km) Scientific Notation Earth 12,700 1.27 × 10 4 km Moon 3480 3.48 × 10 3 km Sun 1,390,000 1.39 × 10 6 km Jupiter 134,000 1.34 × 10 5 km Mercury 4800 4.8 × 10 3 km 10 4 . 22 37, 600, 000, 000, 000, 000, 000, 000 3.76 10 atoms = ´ 10 5 . 20 602, 000, 000, 000, 000, 000, 000 6.02 10 atoms = ´ 10 6 . 20 0.0000000000000000000266 2.66 10 kg - = ´ 10 7 . 21 0.00000000000000000000167 1.67 10 kg - = ´ 10 8 . 8 380,000,000 3.8 10 m = ´ 10 9 . 24 5, 980, 000, 000, 000, 000, 000, 000, 000 5.98 10 kg = ´ 110. 30 2, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 2.0 10 kg = ´ P.2 Beyond the Basics 111. 2 32 x = a. 2 2 2 2 2 32 4 128 x x + = × = × = b. 1 1 1 2 2 2 32 16 2 x x - - = × = × = 112. 3 81 x = a. 1 1 3 3 3 81 3 243 x x + = × = × = b. 2 2 2 1 1 3 3 3 81 81 9 3 9 x x - - æ ö = × = × = × = ç ÷ è ø 11 3 . 5 11 x = a. 1 1 5 5 5 11 5 55 x x + = × = × = b. 2 2 2 1 1 11 5 5 5 11 11 5 25 25 x x - - æ ö = × = × = × = ç ÷ è ø 11 4 . x a b = a. 2 2 2 2 x x a a a b a a b + = × = × = b. 1 1 1 x x b a a a b a a - - = × = × = 115. ( ) ( ) ( ) 2 2 2 2 2 2 2 3 3 2 2 2 2 3 3 2 3 2 3 3 2 2 3 3 3 9 3 3 3 3 3 3 3 3 1 1 3 9 3 n n n n n n n n n n n n n - - - - - - - - - - × × = × = = = = = = 116. ( ) ( ) ( ) ( ) 1 2 2 1 1 2 2 1 1 2 2 1 1 2 3 5 6 6 10 15 2 2 3 3 5 5 5 2 3 2 3 2 5 3 5 2 2 3 3 5 5 5 2 3 2 3 2 5 3 5 2 m m n m n n m n m n m m n m n m n m m m n m n n n m m n n m m m + - + + + - + - + + × × × × × × × × × × × × × = × × × × × × × × × × × × × = × × × × × = 1 2 × 2 3 3 5 m n m - × × × 5 n × 2 5 2 n × × 3 2 n m × 3 2 m n × × 1 2 × 5 n × 1 5 3 5 m m × × × 2 2 2 2 1 2 1 3 5 3 5 5 3 3 5 3 5 m m m m m × × = = = × × ×

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    Section P. 3 Polynomials 13 117. ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 z x y z y xy xz yz x xy yz xz y x z xy xz xz xy yz yz yz xz xz yz - - - - æ ö ¸ = ¸ ç ÷ è ø = × = × = 118. 1 1 1 1 1 1 1 1 1 1 x y z xy yz xz y z x y z x x y z a a a a a a a a a a a a æ ö æ ö æ ö × × ç ÷ ç ÷ ç ÷ è ø è ø è ø = × × = P.2 Getting Ready for the Next Section 119. a. 2 5 2 5 7 x x x x + × = = b. ( ) ( ) 2 3 2 5 10 x x x - = - c. ( )( )( ) 2 3 5 2 3 5 10 2 3 4 24 24 y y y y y + + = = 120. a. 2 2 2 2 5 7 x x x + = b. 2 2 2 3 4 x x x - = - c. 3 3 3 3 3 5 11 9 x x x x - + = 121. False. 2 5 x and 3 3 x are not like terms, so they cannot be combined. 122. ( ) 2 3 2 3 2 1 2 5 3 2 2 5 3 4 10 6 8 10 6 8 x x x x x x x x x + + - + = - + = - + P.3 Polynomials P.3 Practice Problems 1. ( ) ( ) 2 16 7 15 7 889 ft + = 2. ( ) ( ) 3 2 3 2 3 2 7 2 5 2 3 2 1 5 5 2 4 + - + - + + + = + + - x x x x x x x x 3. ( ) ( ) 4 3 2 4 2 4 3 2 3 5 2 7 2 3 5 5 5 12 - + + - - + + - = - - - + x x x x x x x x x x 4. ( ) 3 2 5 4 3 2 4 2 5 8 4 10 - + - = - - + x x x x x x 5. ( )( ) ( ) ( ) 2 2 2 2 2 4 3 2 3 2 4 3 2 5 2 2 7 5 2 7 2 2 7 10 5 35 4 2 14 10 33 14 + - + - = - + - + - + - = - + - - + - = - + - - x x x x x x x x x x x x x x x x x x x x 6. a. ( )( ) 2 2 4 1 7 4 28 7 4 27 7 - + = + - - = + - x x x x x x x b. ( )( ) 2 2 3 2 2 5 6 15 4 10 6 19 10 - - = - - + = - + x x x x x x x 7. ( ) ( ) ( )( ) 2 2 2 2 3 2 3 2 3 2 2 9 12 4 + = + + = + + x x x x x 8. ( )( ) ( ) 2 2 2 1 2 1 2 1 2 1 4 x x x x - + = - = - 9. a. ( )( ) ( ) 2 2 2 2 2 2 2 4 x y x y x y x y + - = - = - b. ( ) ( ) ( ) ( ) ( ) 3 3 2 2 3 3 2 2 3 3 2 2 3 2 2 3 2 3 2 8 3 4 6 8 12 6 x y x x x y y x x xy y x x xy y - = - + - = - + - = - + - P.3 Concepts and Vocabulary 1. The polynomial 7 2 3 2 9 4 - + - + x x x has leading coefficient −3 and degree 7 . 2. When a polynomial is written so that the exponents in each term decrease from left to right, it is said to be in standard form. 3. When a polynomial in x of degree 3 is added to a polynomial in x of degree 4, the resulting polynomial has degree 4 . 4. When a polynomial in x of degree 3 is multiplied by a polynomial in x of degree 4, the resulting polynomial has degree 7 . 5. True 6. True. This is true if A or B or both are zero. 7. False. This is not a polynomial because the term 3 x - does not have an exponent that is either a positive integer or zero. 8. True P.3 Building Skills 9 . A polynomial; 2 2 1 + + x x 10 . N ot a polynomial 11 . N ot a polynomial

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    14 Chapter P Basic Concepts of Algebra 1 2 . A polynomial; 7 5 4 3 3 2 1 + + - + x x x x 1 3 . N ot a polynomial 1 4 . N ot a polynomial 1 5 . A polynomial; in standard form 1 6 . A polynomial; in standard form 17 . D egree: 1; terms: 7 x , 3 18 . D egree: 2; terms: 2 3 , 7 - x 19 . D egree: 4; terms: 4 2 , , 2 , 9 - - x x x 20 . D egree: 7; terms: 7 3 9 , 2 , , 21 - x x x 21 . ( ) ( ) ( ) ( ) ( ) ( ) 3 2 3 3 3 2 2 2 5 3 2 4 2 5 2 3 4 2 3 1 + - + + - + - = + - + + - + + - = - - x x x x x x x x x x x x 22 . ( ) ( ) ( ) ( ) ( ) 3 3 2 3 3 2 3 2 3 1 3 3 1 3 2 2 2 x x x x x x x x x x x x x - + + - + - = + - + - + + - = - - - 23 . ( ) ( ) ( ) ( ) ( ) ( ) 3 2 3 3 3 2 3 2 2 5 4 3 2 4 5 3 5 8 x x x x x x x x x x x x x - + - - - + = - - + - - + - - = - + - 24 . ( ) ( ) ( ) ( ) ( ) ( ) 3 3 2 3 3 2 3 2 2 4 3 7 2 3 2 7 4 2 2 3 9 6 x x x x x x x x x x x x x - + - - + - + = - - - + - - + - - = - - + - 25 . ( ) ( ) ( ) ( ) ( ) ( ) 4 2 4 3 2 4 4 3 2 2 4 3 2 2 3 7 8 6 9 17 2 8 7 6 3 9 17 10 6 12 7 17 x x x x x x x x x x x x x x x x - + - - + - - = - - - - + - - - - = - - + - + 26 . ( ) ( ) 2 2 2 2 2 3 2 2 2 5 4 3 6 6 10 2 8 13 8 14 x x x x x x x x x x - + + - + = - + + - + = - + 27 . ( ) ( ) 2 2 2 2 2 2 3 1 6 3 2 2 6 2 2 18 12 12 24 14 14 x x x x x x x x x x - + + + - - - = - - - - - - = - - - 28 . ( ) ( ) 2 2 2 2 2 2 5 3 4 3 7 1 10 2 6 12 28 4 2 30 2 x x x x x x x x x x - + - + + = - + - - - = - - + 29 . ( ) ( ) ( ) 3 3 2 3 3 2 3 2 3 4 2 2 1 4 3 4 2 2 1 4 2 2 1 y y y y y y y y y y y y y - + + + - - + = - + + + - + - = + - - 30 . ( ) ( ) ( ) 2 2 2 2 2 2 2 5 3 1 2 3 2 5 5 3 1 2 3 2 5 6 6 1 y y y y y y y y y y y y y y + - - - + + + + = + - - + - + + + = + + 31 . 2 6 (2 3) 12 18 + = + x x x x 32 . 2 7 (3 4) 21 28 - = - x x x x 33 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 2 1 2 2 2 2 1 2 2 2 2 2 2 3 4 2 x x x x x x x x x x x x x x x x + + + = + + + + + = + + + + + = + + + 34 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 2 5 2 3 1 2 3 1 5 2 3 1 2 3 10 15 5 2 13 16 5 x x x x x x x x x x x x x x x x - - + = - + - - + = - + - + - = - + - 35 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 2 3 2 1 3 1 2 1 3 3 3 2 2 2 3 5 5 2 x x x x x x x x x x x x x x x x - - + = - + - - + = - + - + - = - + - 36 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 2 2 1 3 4 2 3 4 1 3 4 2 6 8 3 4 2 5 5 4 x x x x x x x x x x x x x x x x + - + = - + + - + = - + + - + = - + + 37 . 2 2 ( 1)( 2) ( 2) 1( 2) 2 2 3 2 + + = + + + = + + + = + + x x x x x x x x x x 38 . 2 2 ( 2)( 3) ( 3) 2( 3) 3 2 6 5 6 + + = + + + = + + + = + + x x x x x x x x x x 39 . 2 2 (3 2)(3 1) 9 3 6 2 9 9 2 + + = + + + = + + x x x x x x x 40 . 2 2 ( 3)(2 5) 2 5 6 15 2 11 15 + + = + + + = + + x x x x x x x 41 . 2 2 ( 4 5)( 3) 4 12 5 15 4 7 15 - + + = - - + + = - - + x x x x x x x

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    Section P. 3 Polynomials 15 42 . 2 2 ( 2 1)( 5) 2 10 5 2 11 5 - + - = - + + - = - + - x x x x x x x 43 . 2 2 (3 2)(2 1) 6 3 4 2 6 7 2 - - = - - + = - + x x x x x x x 44 . 2 2 ( 1)(5 3) 5 3 5 3 5 8 3 - - = - - + = - + x x x x x x x 45 . 2 2 2 2 (2 3 )(2 5 ) 4 10 6 15 4 4 15 - + = + - - = + - x a x a x ax ax a x ax a 46 . 2 2 2 2 (5 2 )( 5 ) 5 25 2 10 5 23 10 - + = + - - = + - x a x a x ax ax a x ax a 47 . 2 2 2 2 ( 2) 4 4 4 4 + - = + + - = + x x x x x x 48 . 2 2 2 2 ( 3) 6 9 6 9 - - = - + - = - + x x x x x x 49 . 2 2 (4 1) 16 8 1 + = + + x x x 50 . 2 2 (3 2) 9 12 4 + = + + x x x 51 . 2 2 2 2 3 3 3 2 4 4 4 3 9 2 16 x x x x x æ ö æ ö æ ö + = + + ç ÷ ç ÷ ç ÷ è ø è ø è ø = + + 52 . 2 2 2 2 2 2 2 2 5 5 5 4 4 5 25 æ ö æ ö æ ö + = + + ç ÷ ç ÷ ç ÷ è ø è ø è ø = + + x x x x x 53. ( ) ( ) ( )( ) ( ) 2 2 2 2 2 3 4 3 2 3 4 4 9 24 16 x y x x y y x xy y - = - + = - + 54. ( ) ( ) ( )( ) ( ) 2 2 2 2 2 2 5 2 2 2 5 5 4 20 25 x y x x y y x xy y + = + + = + + 55. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 4 2 2 2 2 2 4 4 x x x x x + = + + = + + 56. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 4 3 3 2 3 9 6 x x x x x - = - + = - + 57. 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 4 4 x x x x x x x x x x æ ö æ ö æ öæ ö æ ö + = + + ç ÷ ç ÷ ç ÷ç ÷ ç ÷ è ø è ø è øè ø è ø = + + = + + 58. 2 2 2 2 2 2 2 3 3 3 2 3 3 3 9 9 2 2 9 9 y y y y y y y y y y æ ö æ ö æ öæ ö æ ö - = - + ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø è ø è ø è ø = - + = + - 59. ( ) ( ) ( ) 3 2 3 2 3 3 2 2 3 2 3 2 2 6 12 8 x x x x x x x + = + + + = + + + 60. ( ) ( ) ( ) ( ) ( )( ) 3 3 2 3 3 2 2 1 2 3 2 1 3 2 1 1 8 12 6 1 x x x x x x x - = - + - = - + - 61. ( ) 3 3 3 x x + - Note that this is the difference of cubes, with A = x + 3 and B = x . ( ) ( ) ( ) ( ) ( ) ( ) 3 3 2 2 2 2 2 2 2 3 3 3 3 3 6 9 3 3 3 9 9 9 27 27 x x x x x x x x x x x x x x x x x + - é ù = é + - ù + + + + ë û ë û = + + + + + = + + = + + 62. ( ) 3 3 2 x x - - Note that this is the difference of cubes, with A = x – 2 and B = x . ( ) ( ) ( ) ( ) ( ) ( ) 3 3 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 3 6 4 6 12 8 x x x x x x x x x x x x x x x x x - - é ù = é - - ù - + - + ë û ë û = - - + + - + = - - + = - + - 63. ( ) ( ) ( ) 3 2 3 2 3 3 2 2 2 3 2 3 2 2 6 12 8 x y x x y x y x x y xy + = + + + = + + + 64. ( ) ( ) ( ) ( ) ( )( ) ( ) 3 3 2 2 3 3 2 2 3 2 3 2 3 2 3 3 2 3 3 8 36 54 27 x y x x y x y y x x y xy y + = + + + = + + + 65. ( )( ) ( ) 2 2 2 5 2 5 2 5 2 25 4 x x x x + - = - = - 66. ( )( ) ( ) 2 2 2 3 4 3 4 3 4 9 16 x x x x - + = - = - 67. ( )( ) ( ) ( ) 2 2 2 2 2 3 2 3 2 3 4 9 x y x y x y x y + - = - = - 68. ( )( ) ( ) ( ) 2 2 2 2 5 2 5 2 5 2 25 4 x y x y x y x y - + = - = -

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    16 Chapter P Basic Concepts of Algebra 69. 2 2 2 2 1 1 1 1 x x x x x x x x æ öæ ö æ ö + - = - = - ç ÷ç ÷ ç ÷ è øè ø è ø 70. 2 2 2 2 2 2 2 2 2 2 4 4 y y y y y y y y æ öæ ö æ ö æ ö - + = - ç ÷ ç ÷ç ÷ ç ÷ è ø è øè ø è ø = - 71. ( )( ) ( ) 2 2 2 2 2 4 3 3 3 9 x x x x + - = - = - 72. ( )( ) ( ) 2 3 3 3 2 6 2 2 2 4 x x x x - + = - = - 73 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 2 2 3 3 5 2 3 5 3 3 5 2 6 10 3 9 15 2 9 19 15 x x x x x x x x x x x x x x x x - - + = - + - - + = - + - + - = - + - 74 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 2 2 4 3 4 3 2 4 3 4 3 2 8 6 6 5 6 x x x x x x x x x x x x x x x x - - - = - - - - - = - - - + + = - + + 75 . ( ) ( ) ( ) ( ) 2 2 2 2 2 3 3 1 1 1 1 1 1 1 y y y y y y y y y y y y y y + - + = - + + - + = - + + - + = + 76 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 4 4 16 4 16 4 4 16 4 16 4 16 64 64 y y y y y y y y y y y y y y + - + = - + + - + = - + + - + = + 77 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 6 6 36 6 36 6 6 36 6 36 6 36 216 216 x x x x x x x x x x x x x x - + + = + + - + + = + + - - - = - 78 . ( ) ( ) ( ) ( ) 2 2 2 3 2 2 3 1 1 1 1 1 1 1 x x x x x x x x x x x x x x - + + = + + - + + = + + - - - = - 79 . 2 2 2 2 ( 2 )(3 5 ) 3 5 6 10 3 11 10 + + = + + + = + + x y x y x xy xy y x xy y 80 . 2 2 2 2 (2 )(7 2 ) 14 4 7 2 14 11 2 + + = + + + = + + x y x y x xy xy y x xy y 81 . 2 2 2 2 (2 )(3 7 ) 6 14 3 7 6 11 7 - + = + - - = + - x y x y x xy xy y x xy y 82 . 2 2 2 2 ( 3 )(2 5 ) 2 5 6 15 2 15 - + = + - - = - - x y x y x xy xy y x xy y 83 . ( ) ( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 4 2 2 4 2 x y x y x y x y x y x y x y x y x y x y x y x y x x y y - + = - - + + = é - + ùé - + ù ë ûë û = - - = - + 84 . ( ) ( ) ( )( )( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 4 2 2 4 2 2 2 2 2 2 2 2 2 2 4 4 16 8 x y x y x y x y x y x y x y x y x y x y x y x y x x y y + - = + + - - = é - + ùé - + ù ë ûë û = - - = - + 85 . ( )( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 3 2 2 2 2 3 3 2 3 2 4 4 4 4 4 4 4 4 4 4 3 4 x y x y x y x xy y x x xy y y x xy y x x y xy x y xy y x x y y + - = + - + = - + + - + = - + + - + = - + 86 . ( )( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 3 2 2 2 2 3 3 2 3 2 4 4 4 4 4 4 4 4 4 4 3 4 x y x y x y x xy y x x xy y y x xy y x x y xy x y xy y x x y y - + = - + + = + + - + + = + + - - - = + - 87 . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 3 3 2 3 2 2 3 3 2 2 3 3 2 2 3 4 3 2 2 3 3 2 2 3 4 4 3 3 4 2 2 3 2 3 2 2 2 6 12 8 2 6 12 8 2 6 12 8 6 12 8 2 12 24 16 4 16 16 x y x y x x y x y y x y x x y xy y x y x x x y xy y y x x y xy y x x y x y xy x y x y xy y x x y xy y - + = + - + - + - × + = - + - + = - + - + - + - = - + - + - + - = - + -

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    Section P. 3 Polynomials 17 88 . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 2 2 3 3 2 2 3 2 2 2 3 2 3 2 2 8 12 6 2 x y x y x x y x y y x y x x y xy y x y + - é ù = + + + ë û × - = + + + - ( ) ( ) 3 2 2 3 3 2 2 3 4 3 2 2 3 3 2 2 3 4 4 3 3 4 2 8 12 6 8 12 6 16 24 12 2 8 12 6 16 16 4 x x x y xy y y x x y xy y x x y x y xy x y x y xy y x x y xy y = + + + - + + + = + + + - - - - = + - - For exercises 89 − 94 , use ( ) ( ) 2 2 2 2 2 2 . a b a b ab a b ab + = + - = - + 89 . ( ) 2 2 2 2 4, 3 2 4 2 3 16 6 10 x y xy x y x y xy + = = + = + - = - × = - = 90 . ( ) 2 2 2 2 3, 2 2 3 2 2 9 4 13 x y xy x y x y xy - = = + = - + = + × = + = 91 . a. 1 3 x x + = Let a = x and 1 . b x = Then 1 1. ab x x æ ö = = ç ÷ è ø 2 2 2 2 1 1 1 2 3 2 9 2 7 x x x x x x æ ö æ ö + = + - ç ÷ ç ÷ è ø è ø = - = - = b. Let 2 a x = and 2 1 . b x = Using the result from part a, we have 2 4 2 2 4 2 2 2 1 1 1 2 7 2 49 2 47 x x x x x x æ ö æ ö + = + - ç ÷ ç ÷ è ø è ø = - = - = 92 . a. 1 2 x x - = Let a = x and 1 . b x = Then 1 1. ab x x æ ö = = ç ÷ è ø 2 2 2 2 1 1 1 2 2 2 4 2 6 x x x x x x æ ö æ ö + = - + ç ÷ ç ÷ è ø è ø = + = + = b. Let 2 a x = and 2 1 . b x = Using the result from part a, we have 2 4 2 2 4 2 2 2 1 1 1 2 6 2 36 2 34 x x x x x x æ ö æ ö + = + - ç ÷ ç ÷ è ø è ø = - = - = 93 . Let a = 3 x and b = 2 y . Then ab = 6 xy . ( ) ( ) 2 2 2 2 9 4 3 2 2 6 12 2 6 6 144 72 72 x y x y xy + = + - = - × × = - = 94 . Let a = 3 x and b = 7 y . Then ab = 21 xy . ( ) ( ) ( )( ) 2 2 2 2 9 49 3 7 2 21 10 2 21 1 100 42 58 x y x y xy + = - + = + - = - = P.3 Applying the Concepts 95 . ( ) ( ) 2 0.025 6 0.44 6 4.28 $6.02 - + + = in 2012 (six years after 2006) 96 . ( ) ( ) 2 0.035 4 0.15 4 5.17 6.33 + + = In 2012 (four years after 2008), theater grosses were 6.33 billion dollars. 97 . ( ) 2 0.1 40 40 50 $250.00 + + = 98 . ( ) ( ) 2 2 15 10 5 15 5 75 25 75 $100.00 - + = + = + = 99 . ( ) ( ) ( ) 2 16 5 20 5 16 25 100 500 feet d = + = + = 100 . ( ) ( ) ( ) 2 16 2 10 2 16 4 20 84 feet d = + = + = 101 . a. 22.50 - + x b. ( )( ) ( ) 2 2 2 30 10 22.50 30 22.50 30 225 10 675 195 10 10 195 675 x x x x x x x x x + - = - + - = + - = - + + 102 . a. 10 250 + n b. 2 2 (50 2 )(10 250) 50(10 ) 50(250) 2 (10 ) 2 (250) 500 12, 500 20 500 20 12,500 n n n n n n n n n n - + = + - - = + - - = - +

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    18 Chapter P Basic Concepts of Algebra P.3 Beyond the Basics 103. ( ) ( )( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 a b c a b c a b c a a b c b a b c c a b c a ab ac ab b bc ac bc c a b c ab bc ac + + = + + + + = + + + + + + + + = + + + + + + + + = + + + + + 104. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 4 2 3 2 4 3 2 1 1 2 2 1 2 1 1 2 2 2 2 3 2 1 x x x x x x x x x x x x x x x x x + + = + + + + + = + + + + + = + + + + 105. ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 4 x y z x y z x y y z x z x y z xy yz xz + - = + + - + + - + - = + + + - - ( ) ( ) ( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 4 x y z x y z x y y z x z x y z xy yz xz - + = + - + + - + - + = + + - - + ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 4 4 2 4 4 4 2 4 8 8 x y z x y z x y z xy yz xz x y z xy yz xz xy xz + - - - + = + + + - - - + + - - + = - Alternatively, recognize the difference of two squares. ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 2 4 2 2 8 8 x y z x y z x y z x y z x y z x y z x y z xy xz + - - - + = é + - + - + ùé + - - - + ù ë ûë û = - = - 106. 8; 12 a b c ab bc ac + + = + + = ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8 2 12 64 24 40 a b c a b c ab bc ac a b c ab bc ac a b c a b c a b c + + = + + + + + = + + + + + = + + + = + + + = + + 107. 2 2 2 12; 44 x y z x y z + + = + + = ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 12 44 2 144 44 2 100 2 50 x y z x y z xy yz xz x y z xy yz xz xy yz xz xy yz xz xy yz xz xy yz xz + + = + + + + + = + + + + + = + + + = + + + = + + = + + 108. 2 2 2 64; 18 x y z xy yz xz + + = + + = ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 64 2 18 100 100 10 x y z x y z xy yz xz x y z xy yz xz x y z x y z x y z + + = + + + + + = + + + + + + + = + + + = + + = = 109. ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 2 2 3 3 3 3 a b c a b c ab bc ca a a b c ab bc ca b a b c ab bc ca c a b c ab bc ca a ab ac a b abc ca a b b bc ab b c abc a c b c c abc bc c a a b c abc + + + + - - - = + + - - - + + + - - - + + + - - - = + + - - - + + + - - - + + + - - - = + + -

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    Section P. 4 Factoring Polynomials 19 110. From exercise 109, ( ) ( ) 2 2 2 3 3 3 3 3 3 3 3 3 0 3 0 3 3 a b c ab bc ca a b c abc a b c abc a b c abc + + - - - = + + - Þ = + + - Þ + + = 111. Let a = x – y , b = y – z , and c = z – x . Then ( ) ( ) ( ) 0. a b c x y y z z x + + = - + - + - = From exercise 110, if 0, a b c + + = then 3 3 3 3 . a b c abc + + = So, ( ) ( ) ( ) ( )( )( ) 3 3 3 3 . x y y z z x x y y z z x - + - + - = - - - 112. Let a = 2 x – 3 y , b = 3 y – 5 z , and c = 5 z – 3 x . Then ( ) ( ) ( ) 2 3 3 5 5 3 0. a b c x y y z z x + + = - + - + - = From exercise 110, if 0, a b c + + = then 3 3 3 3 . a b c abc + + = So, ( ) ( ) ( ) ( )( )( ) 3 3 3 2 3 3 5 5 2 3 2 3 3 5 5 2 . x y y z z x x y y z z x - + - + - = - - - 113. From exercise 109, ( ) ( ) 2 2 2 3 3 3 3 . a b c a b c ab bc ca a b c abc + + + + - - - = + + - If 8 a b c + + = and 19, ab bc ca + + = we have ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 3 3 3 2 2 2 3 3 3 3 8 19 3 (1) a b c a b c ab bc ca a b c a b c ab bc ca a b c abc a b c a b c abc + + + + - - - = + + + + - + + = + + - Þ + + - = + + - From exercise 103, ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8 2 19 64 38 26 a b c a b c ab bc ac a b c ab bc ca a b c a b c a b c + + = + + + + + = + + + + + Þ = + + + Þ = + + + Þ + + = Substituting into equation (1) gives us ( ) 3 3 3 3 3 3 8 26 19 3 56 3 . a b c abc a b c abc - = + + - Þ = + + - 114. From exercise 109, ( ) ( ) 2 2 2 3 3 3 3 . a b c a b c ab bc ca a b c abc + + + + - - - = + + - If 9 a b c + + = and 11, ab bc ca + + = we have ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 3 3 3 2 2 2 3 3 3 3 9 11 3 (1) a b c a b c ab bc ca a b c a b c ab bc ca a b c abc a b c a b c abc + + + + - - - = + + + + - + + = + + - Þ + + - = + + - From exercise 103, ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 9 2 11 81 22 59 a b c a b c ab bc ac a b c ab bc ca a b c a b c a b c + + = + + + + + = + + + + + Þ = + + + Þ = + + + Þ + + = Substituting into equation (1) gives us ( ) 3 3 3 3 3 3 9 59 11 3 432 3 . a b c abc a b c abc - = + + - Þ = + + - P.3 Getting Ready for the Next Section a b a + b ab 115. 3 4 7 12 116. – 3 5 2 – 15 117. 4 2 6 8 118. 3 – 5 – 2 – 15 119. 2 5 7 10 a b a + b ab 120. 3 5 8 15 121. – 5 7 2 – 35 122. 5 – 7 – 2 – 35 123. – 2 – 3 – 5 6 124. 2 3 5 6 P.4 Factoring Polynomials P.4 Practice Problems 1. a. ( ) 5 3 3 2 6 14 2 3 7 + = + x x x x b. ( ) 5 4 2 2 3 2 7 21 35 7 3 5 + + = + + x x x x x x c. ( ) ( ) ( ) ( ) 2 2 5 2 5 2 x x y x y x y x - + - = - + 2. a ( )( ) 2 6 8 4 2 + + = + + x x x x

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    20 Chapter P Basic Concepts of Algebra b. ( )( ) 2 3 10 5 2 - - = - + x x x x 3. a. ( ) 2 2 4 4 2 x x x + + = + b. ( ) 2 2 9 6 1 3 1 x x x - + = - 4. a. ( )( ) 2 16 4 4 - = - + x x x b. ( )( ) 2 4 25 2 5 2 5 - = - + x x x 5. ( )( ) ( )( ) ( ) 4 2 2 2 81 9 9 3 3 9 x x x x x x - = - + = - + + 6. a. ( ) ( ) 3 2 125 5 5 25 x x x x - = - + + b. ( ) ( ) 3 2 27 8 3 2 9 6 4 + = + - + x x x x 7. Following the reasoning in Example 7, in four years, the company will have invested the intial 12 million dollars, plus an additional 4 million dollars. Thus, the total investment is 16 million dollars. To find the profit or loss with 16 million dollars invested, let x = 16 in the profit - loss polynomial: ( )( ) ( ) ( )( ) ( ) 2 2 0.012 14 14 196 0.012 16 14 16 14 16 196 16.224 - + + = - + × + = x x x The company will have made a profit of 16.224 million dollars in four years. 8 . a. ( ) ( ) ( ) ( ) 3 2 2 2 3 3 3 1 3 3 1 x x x x x x x x + + + = + + + = + + b. ( ) ( ) ( ) ( ) ( )( )( ) 3 2 2 2 28 20 7 5 4 7 5 7 5 7 5 4 1 7 5 2 1 2 1 x x x x x x x x x x x - - + = - - - = - - = - + - c. ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 1 2 1 1 1 1 1 1 - - - = - + + = - + = - + + + = - - + + x y y x y y x y x y x y x y x y 9. ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 5 9 6 9 6 9 3 3 3 3 3 x x x x x x x x x x x x x x x x x x + + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - + 10 . a. ( ) ( ) ( ) ( ) ( )( ) 2 2 2 5 11 2 5 10 2 5 10 2 5 2 1 2 5 1 2 x x x x x x x x x x x x x + + = + + + = + + + = + + + = + + b. ( ) ( ) ( ) ( ) ( )( ) 2 2 2 4 3 4 4 3 3 4 4 3 3 4 1 3 1 4 3 1 x x x x x x x x x x x x x + - = + - - = + - + = + - + = - + P.4 Concepts and Vocabulary 1. The polynomials x + 2 and x − 2 are called factors of the polynomial 2 4. - x 2. The polynomial 3 y is the greatest common monomial factor of the polynomial 2 3 6 . + y y 3. The GCF of the polynomial 3 2 10 30 + x x is 2 10 . x 4. A polynomial that cannot be factored as a product of two polynomials (excluding constant polynomials 1) ± is said to be irreducible . 5. True 6. True 7. False. The polynomial 2 4 x - can be factored as ( )( ) 2 2 . x x + - Therefore, it is not irreducible. 8. True P.4 Building Skills 9 . 8 24 8( 3) x x - = - 10 . 5 25 5( 5) x x + = + 11 . 2 6 12 6 ( 2) x x x x - + = - - 12 . 2 2 3 21 3( 7) x x - + = - -

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    Section P. 4 Factoring Polynomials 21 13 . 2 3 2 7 14 7 (1 2 ) x x x x + = + 14 . 3 4 3 9 18 9 (1 2 ) x x x x - = - 15 . ( ) 4 3 2 2 2 2 2 1 x x x x x x + + = + + 16 . ( ) 4 3 2 2 2 5 7 5 7 x x x x x x - + = - + 1 7 . 3 2 2 3 (3 1) x x x x - = - 1 8 . 3 2 2 2 2 2 ( 1) x x x x + = + 1 9 . 3 2 2 8 4 4 (2 1) ax ax ax x + = + 20 . ( ) 4 2 3 2 2 1 ax ax ax ax x x - + = - + 21. ( ) ( ) ( )( ) 3 3 x x y x y x x y - + - = + - 22. ( ) ( ) ( )( ) 2 1 3 1 2 3 1 a a a a a + - + = - + 23. ( ) ( ) ( )( ) 3 2 5 2 3 5 2 x x y y x y x y x y + + + = + + 24. ( ) ( ) ( )( ) 2 5 3 7 5 3 2 7 5 3 x x y y x y x y x y - + - = + - 25. ( ) ( ) ( )( ) ( )( ) 2 2 3 2 2 2 3 2 1 y y y y y y + - + = + + - = + - 26. ( ) ( ) ( ) ( ) ( ) ( )( ) 2 1 1 1 1 1 1 2 y y y y y y + + + = + + + = + + 27 . ( ) ( ) ( ) ( ) 3 2 2 2 3 3 3 1 3 3 1 x x x x x x x x + + + = + + + = + + 28 . ( ) ( ) ( ) ( ) 3 2 2 2 5 5 5 1 5 5 1 x x x x x x x x + + + = + + + = + + 29 . ( ) ( ) ( ) ( ) 3 2 2 2 5 5 5 1 5 5 1 x x x x x x x x - + - = - + - = - + 30 . ( ) ( ) ( ) ( ) 3 2 2 2 7 7 7 1 7 7 1 x x x x x x x x - + - = - + - = - + 31 . ( ) ( ) ( ) ( ) 3 2 2 2 6 4 3 2 2 3 2 1 3 2 3 2 2 1 x x x x x x x x + + + = + + + = + + 32 . ( ) ( ) ( ) ( ) 3 2 2 2 3 6 2 3 2 1 2 2 3 1 x x x x x x x x + + + = + + + = + + 33 . ( ) ( ) ( )( ) ( )( ) 7 5 4 2 5 2 2 2 2 5 2 2 2 3 12 4 3 4 3 1 3 1 3 1 4 3 1 4 1 x x x x x x x x x x x x x x + + + = + + + = + + = + + 34 . ( ) ( ) ( )( ) ( )( ) 7 5 4 2 5 2 2 2 2 5 2 2 2 3 3 3 3 1 1 1 3 1 3 1 x x x x x x x x x x x x x x + + + = + + + = + + = + + 35. ( ) ( ) ( ) ( ) ( )( ) xy ab bx ay xy bx ab ay x y b a b y x a b y + + + = + + + = + + + = + + 36. ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 a x b bx a a x a b bx a x b x a b x - - + = + - + = + - + = - + 37 . 2 7 12 ( 3)( 4) x x x x + + = + + 38 . 2 8 15 ( 3)( 5) x x x x + + = + + 39 . 2 6 8 ( 4)( 2) x x x x - + = - - 40 . 2 9 14 ( 7)( 2) x x x x - + = - - 41 . 2 3 4 ( 4)( 1) x x x x - - = - + 42 . 2 5 6 ( 6)( 1) x x x x - - = - + 43 . I rreducible 44 . I rreducible 4 5. 2 2 36 (2 9)( 4) x x x x + - = + - 4 6. 2 2 3 27 (2 9)( 3) x x x x + - = + - 4 7. 2 6 17 12 (2 3)(3 4) x x x x + + = + + 4 8. 2 8 10 3 (2 3)(4 1) x x x x - - = - + 4 9. 2 3 11 4 (3 1)( 4) x x x x - - = + - 5 0. 2 5 7 2 (5 2)( 1) x x x x + + = + + 5 1. I rreducible 5 2. I rreducible 53. ( )( ) 2 2 20 4 5 x xy y x y x y - - = + -

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    22 Chapter P Basic Concepts of Algebra 54. ( )( ) 2 2 8 2 15 4 5 2 3 p pq q p q p q - - = + - 55. ( )( ) 2 2 15 28 15 28 3 4 5 7 x x x x x x - + = + - = - + 56. ( )( ) 2 2 24 35 4 4 24 35 2 7 2 5 y y y y y y + + = + + = + + 57 . 2 2 6 9 ( 3) x x x + + = + 58 . 2 2 8 16 ( 4) x x x + + = + 59 . 2 2 9 6 1 (3 1) x x x + + = + 60 . 2 2 36 12 1 (6 1) x x x + + = + 61 . 2 2 25 20 4 (5 2) x x x - + = - 62 . 2 2 2 64 32 4 4(16 8 1) 4(4 1) x x x x x + + = + + = + 63 . 2 2 49 42 9 (7 3) x x x + + = + 64 . 2 2 9 24 16 (3 4) x x x + + = + 65 . 2 64 ( 8)( 8) x x x - = - + 66 . 2 121 ( 11)( 11) x x x - = - + 67 . 2 4 1 (2 1)(2 1) x x x - = - + 68 . 2 9 1 (3 1)(3 1) x x x - = - + 69 . 2 16 9 (4 3)(4 3) x x x - = - + 70 . 2 25 49 (5 7)(5 7) x x x - = - + 71 . ( )( ) ( )( ) ( ) 4 2 2 2 1 1 1 1 1 1 x x x x x x - = - + = - + + 72 . ( )( ) ( )( ) ( ) 4 2 2 2 81 9 9 3 3 9 x x x x x x - = - + = - + + 73 . ( ) ( )( ) 4 4 2 2 20 5 5 4 1 5 2 1 2 1 x x x x - = - = - + 74 . ( ) ( )( ) 4 4 2 2 12 75 3 4 25 3 2 5 2 5 x x x x - = - = - + 75 . ( ) ( ) 3 3 3 2 64 4 4 4 16 x x x x x + = + = + - + 76 . ( ) ( ) 3 3 3 2 125 5 5 5 25 x x x x x + = + = + - + 77 . ( ) ( ) 3 3 3 2 27 3 3 3 9 x x x x x - = - = - + + 78 . ( ) ( ) 3 3 3 2 216 6 6 6 36 x x x x x - = - = - + + 79 . ( ) ( ) 3 3 3 2 8 2 2 4 2 x x x x x - = - = - + + 80 . ( ) ( ) 3 3 3 2 27 3 3 9 3 x x x x x - = - = - + + 81 . ( ) ( ) 3 3 3 2 8 27 (2 ) 3 2 3 4 6 9 x x x x x - = - = - + + 82 . ( ) ( ) ( ) 3 3 3 2 8 125 2 5 2 5 4 10 25 x x x x x - = - = - + + 83 . ( ) ( ) ( ) ( ) ( ) 3 3 3 3 2 40 5 5 8 1 5 2 1 5 2 1 4 2 1 x x x x x x + = + = + = + - + 84 . ( ) ( ) ( ) ( ) 3 3 3 3 2 7 56 7 8 7 2 7 2 2 4 x x x x x x + = + = + = + - + 85 . ( ) ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 25 10 9 25 10 25 9 5 9 5 3 5 3 3 5 3 5 x x x x x x x x x x x x x x x x x x + + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - + 86 . ( ) ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 1 2 1 2 1 1 1 1 1 1 x x x x x x x x x x x x x x x x x x + + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - +

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    Section P. 4 Factoring Polynomials 23 87 . ( ) ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 15 64 16 64 16 64 8 8 8 8 8 x x x x x x x x x x x x x x x x x x + + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - + 88 . ( ) ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 7 9 6 9 6 9 3 3 3 3 3 x x x x x x x x x x x x x x x x x x - + = - - + = - + - = - - é ùé ù = - + - - ë ûë û = + - - - 89 . ( ) ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 16 8 9 16 8 16 9 4 9 4 3 4 3 3 4 3 4 x x x x x x x x x x x x x x x x x x - + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - + 90 . ( ) ( ) ( ) ( ) ( )( ) 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2 16 36 12 4 36 12 36 4 6 4 6 2 6 2 2 6 2 6 x x x x x x x x x x x x x x x x x x - + = - - + = - + - = - - é ùé ù = - + - - ë ûë û = + - - - 91 . ( ) ( ) ( ) ( ) ( )( ) 4 4 2 2 4 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 2 4 2 2 2 2 2 2 2 2 x x x x x x x x x x x x x x x x x + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - + 92 . ( ) ( ) ( ) ( ) ( )( ) 4 4 2 2 4 2 2 2 2 2 2 2 2 2 64 16 16 64 16 64 16 8 16 8 4 8 4 4 8 4 8 x x x x x x x x x x x x x x x x x + = + - + = + + - = + - é ùé ù = + + + - ë ûë û = + + - + 93 . 2 1 16 (1 4 )(1 4 ) x x x - = + - 94 . 2 4 25 (2 5 )(2 5 ) x x x - = - + 95 . 2 2 6 9 ( 3) x x x - + = - 96 . 2 2 8 16 ( 4) x x x - + = - 97 . 2 2 4 4 1 (2 1) x x x + + = + 98 . 2 2 16 8 1 (4 1) x x x + + = + 99 . ( ) 2 2 2 8 10 2 4 5 2( 5)( 1) x x x x x x - - = - - = - + 100 . ( ) ( )( ) 2 2 5 10 40 5 2 8 5 4 2 x x x x x x - - = - - = - + 101 . 2 2 3 20 (2 5)( 4) x x x x + - = - + 102 . 2 2 7 30 (2 5)( 6) x x x x - - = + - 103 . 2 24 36 x x - + is irreducible. 104 . 2 20 25 x x - + is irreducible . 105 . 5 4 3 3 2 3 2 3 12 12 3 ( 4 4) 3 ( 2) x x x x x x x x + + = + + = + 106 . 5 4 3 3 2 3 2 2 16 32 2 ( 8 16) 2 ( 4) x x x x x x x x + + = + + = + 107 . 2 9 1 (3 1)(3 1) x x x - = - + 108 . 2 16 25 (4 5)(4 5) x x x - = - + 109 . 2 2 16 24 9 (4 3) x x x + + = + 110 . 2 2 4 20 25 (2 5) x x x + + = + 111 . 2 15 + x is irreducible. 112 . 2 24 + x is irreducible. 113 . ( ) ( )( ) 3 2 2 45 8 4 45 8 4 5 2 9 2 x x x x x x x x x + - = + - = + - 114 . ( ) ( )( ) 3 2 2 25 40 9 25 40 9 5 9 5 1 x x x x x x x x x + - = + - = + - 115 . ( ) ( )( ) 2 2 3 2 2 7 8 7 8 8 ax a x a a x ax a a x a x a - - = - - = - +

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    24 Chapter P Basic Concepts of Algebra 116 . ( ) ( )( ) 2 2 3 2 2 10 24 10 24 12 2 ax a x a a x ax a a x a x a - - = - - = - + 117 . ( ) ( ) ( )( ) 2 2 2 2 2 2 16 8 16 8 16 16 4 16 4 4 4 4 x a x x x a x a x a x a - + + = + + - = + - = + - + + 118 . ( ) ( ) ( )( ) 2 2 2 2 2 2 36 6 9 6 9 36 3 36 3 6 3 6 x a x x x a x a x a x a - + + = + + - = + - = + - + + 119 . ( ) ( ) 5 4 3 2 3 2 2 2 3 3 12 12 3 4 4 3 2 x x y x y x x xy y x x y + + = + + = + 120 . ( ) ( ) 5 4 3 2 3 2 2 2 3 4 24 36 4 6 9 4 3 x x y x y x x xy y x x y + + = + + = + 121 . ( ) ( ) ( )( ) 2 2 2 2 2 2 25 4 4 4 4 25 2 25 2 5 2 5 x a x x x a x a x a x a - + + = + + - = + - = + - + + 122 . ( ) ( ) ( )( ) 2 2 2 2 2 2 16 4 4 4 4 16 2 16 2 4 2 4 x a x x x a x a x a x a - + + = + + - = + - = + - + + 123 . ( ) ( ) 6 5 4 2 4 2 2 2 4 18 12 2 2 9 6 2 3 x x y x y x x xy y x x y + + = + + = + 124 . ( ) ( ) 6 5 4 2 4 2 2 2 4 12 12 3 3 4 4 3 2 x x y x y x x xy y x x y + + = + + = + P.4 Applying the Concepts 125 . If one side of the garden is x feet, and the perimeter is 16 feet, then 16 2 8 2 x x - = - gives the other dimension of the rectangle. So, the area of the garden is (8 ). x x - 126 . If one side of the tray is x inches, and the perimeter is 42 inches, then 42 2 21 2 x x - = - gives the other dimension of the tray. So, the area of the tray is (21 ). x x - Use the figure below for exercises 1 27 and 1 28 . 127 . If x = the length of the cut corner, then 36 – 2 x = the length of the box, and 16 – 2 x = the width of the box. The height of the box is x . So, (36 2 )(16 2 ) (2(18 ))(2(8 )) 4 (18 )(8 ) v x x x x x x x x x = - - = - - = - - 1 28 . 2 2 2 2 2 . . (36 2 )(16 2 ) 2 (36 2 ) 2 (16 2 ) 576 104 4 72 4 32 4 576 4 4(144 ) 4(12 )(12 ) = - - + - + - = - + + - + - = - = - = - + S A x x x x x x x x x x x x x x x x 1 29 . The area of the outside circle = 2 4 cm . p The area of the inside circle = 2 2 cm . x p So the area of the disk = area of outside circle – area of inside circle = 2 2 4 (4 ) x x p p p - = - = 2 (2 )(2 ) cm . x x p - + 1 30 . The volume of the inside cylinder is 2 3 8 p × = 3 72 ft p , and the volume of the outside cylinder is 2 2 3 8 8 ft . x x p p × = So the volume between the cylinders is 2 2 8 72 8 ( 9) x x p p p - = - 3 8 ( 3)( 3) ft . x x p = - +

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    Section P. 4 Factoring Polynomials 25 131. I f one side of the fence is x feet, and the rancher needs a total of 2800 feet of fencing, then the width of the fence is 2800 – 2 x . So, the area of the pen is 2 (2800 2 ) 2800 2 x x x x - = - = 2 2 (1400 ) ft . x x - 1 32 . The area of the figure = the area of the rectangle plus the area of the circle. Find the length of the rectangle as follows: The circumference of the circle = 2 2 x x p p æ ö = ç ÷ è ø . The perimeter of the figure is 48, so the length of the rectangle is 48 . 2 x p - The area of the circle is 2 4 x p and the area of the rectangle is 24 2 x x p æ ö - = ç ÷ è ø 2 24 . 2 x x p - So . the area of the figure is 2 2 2 1 24 (96 ) 4 2 4 x x x x x p p p æ ö + - = - = ç ÷ è ø 2 1 (96 ) in. 4 x x p - P.4 Beyond the Basics 133. ( ) ( ) ( ) ( ) 3 3 2 2 2 2 1 x y x y x y x xy y x y x y x xy y + + + = + - + + + = + - + + 134. ( ) ( )( ) ( )( ) 2 4 2 4 2 2 2 2 2 2 2 2 11 1 2 1 9 1 9 1 3 1 3 3 1 3 1 x x x x x x x x x x x x x x x - + = - + - = - - = - - - + = - - + - 135. ( ) ( )( ) ( )( ) 2 4 4 2 2 4 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x y x y x x y y x y x y x y x y xy x y xy x xy y x xy y + + = + + - = + - = + - + + = - + + + 136. ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) 2 2 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 4 4 4 2 4 2 4 2 2 2 2 2 2 2 2 x y x y x y x y x x y y x y x y x y x y xy x y xy x xy y x xy y + = + + - = + + - = + - = + - + + = - + + + 137. ( ) ( )( ) ( )( )( )( ) ( )( )( )( ) 2 6 3 2 3 3 64 8 8 8 2 2 4 2 2 4 2 2 2 4 2 4 x x x x x x x x x x x x x x x x - = - = - + = - + + + - + = - + + + - + 138. ( )( ) ( ) ( ) ( ) ( ) 6 3 3 3 2 2 26 27 27 1 3 3 9 1 1 x x x x x x x x x x - - = - + = - + + + - +

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    26 Chapter P Basic Concepts of Algebra 139. ( ) ( ) ( ) ( ) ( )( ) ( )( )( )( ) 2 2 2 2 2 2 2 3 38 3 80 3 40 3 2 3 40 3 2 8 5 2 1 x x x x x x x x x x x x x x x x é ùé ù - - - - = - - - + ë ûë û = - - - + = - + - - 140. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 2 2 2 2 4 2 4 2 4 2 4 2 2 2 2 x y z yz x y z yz x y yz z x y z x y z x y z x y z x y z - - + = - + - = - - + = - - = - - + - = - + + - 141. From exercise 109 in Section P.3, we have ( ) ( ) 2 2 2 3 3 3 3 . a b c a b c ab bc ca a b c abc + + + + - - - = + + - Then, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 1 2 a b c abc a b c a b c ab bc ca a b c a b c ab bc ca a b c a b c ab bc ca a b c a ab b b bc c c ca a a b c a b b c c a + + - = + + + + - - - = + + × + + - - - = + + + + - - - = + + - + + - + + - + é ù = + + - + - + - ë û If 3 3 3 3 0, a b c abc + + - = then either 0 a b c + + = or ( ) ( ) ( ) 2 2 2 0 . a b b c c a a b c - + - + - = Þ = = 142. See exercise 111 in section P.3. ( ) ( ) ( ) ( )( )( ) 3 3 3 3 x y y z z x x y y z z x - + - + - = - - - 143. Let ( ) ( ) , , a z x y xz yz b x y z xy xz = - = - = - = - and ( ) . c y z x yz xy = - = - Then 0, a b c xz yz xy xz yz xy + + = - + - + - = and we can apply Section P.3 exercise 110. ( ) ( ) ( ) ( )( )( ) 3 3 3 3 3 3 3 z x y x y z y z x xyz x y y z z x - + - + - = - - - 144. ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 2 2 2 2 2 2 3 3 3 x y y z z x x y y z z x - + - + - - + - + - In the numerator, let 2 2 2 2 , , a x y b y z = - = - and 2 2 . c z x = - Then a + b + c = 0, and we can apply Section P.3 exercise 11 1 . Use the result from exercise 142 above for the denominator. ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )( )( ) ( )( )( )( )( )( ) ( )( )( ) ( )( )( ) 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 x y y z z x x y y z z x x y y z z x x y y z z x x y x y y z y z z x z x x y y z z x x y y z z x - + - + - - - - = - - - - + - + - + - + - + - = - - - = + + + P.4 Getting Ready for the Next Section 145. 1 1 3 2 5 4 6 12 12 12 + = + = 146. 3 1 9 2 7 8 12 24 24 24 - = - = 147. 9 1 20 5 16 × 4 27 3 4 15 = 148. 2 1 5 3 6 × 3 5 9 =

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    Section P. 5 Rational Expressions 27 149. ( ) 2 2 4 6 2 2 3 2 3 x x x x x x = = + + + 15 0 . ( ) ( ) 2 3 2 2 2 3 2 3 2 3 1 1 x x x x x x x x x x + + + = = + + + 15 1 . ( )( ) ( )( ) 2 2 1 2 3 2 2 1 3 3 4 3 x x x x x x x x x x + + + + + = = + + + + + 152. ( )( ) ( )( ) 2 2 3 2 6 3 2 2 2 4 x x x x x x x x x - + - - - = = - + - - P.5 Rational Expressions P.5 Practice Problems 1. ( ) ( ) ( )( ) 0.05 1 0.1 0.32 0.95 0.1 0.05 1 0.1 - » + - The likelihood that a student who tests positive is a nonuser when the test that is used is 95% accurate is about 32%. 2. a. ( ) ( ) 2 3 2 2 2 4 2 8 2 3 4 3 3 12 x x x x x x x x x + + = = + + b. ( )( ) ( ) 2 2 2 2 2 4 2 2 4 4 2 x x x x x x x x - + - - = = + + + + 3. ( )( ) ( ) ( ) ( ) ( ) 2 2 3 2 3 2 4 3 4 3 2 4 3 3 2 3 2 2 3 2 3 4 4 7 28 4 4 7 28 2 8 2 8 2 3 2 8 4 4 7 28 3 1 2 4 4 1 7 4 3 14 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x - - - - + + = ¸ + + + + - - + = × + + - + + = × + + - = 4. a. ( ) ( ) ( )( ) 2 2 2 2 2 10 5 22 5 22 2 20 36 36 36 7 42 36 7 6 7 6 6 6 x x x x x x x x x x x x x + + + + + + = - - - + = - + = = - + - b. ( ) ( ) ( )( ) 2 2 2 2 4 1 3 4 4 1 3 4 12 12 12 3 12 3 4 3 1 4 x x x x x x x x x x x x x x x x x + - + + + - = + - + - + - - = + - - = + - = + 5. a. ( ) ( )( ) ( ) ( )( ) 2 3 2 5 2 5 3 2 2 5 2 5 + + - - + = + + - + - x x x x x x x x x x x x ( )( ) ( )( ) ( ) ( )( ) 2 2 2 2 10 3 6 5 4 2 5 2 5 5 4 2 5 - + + - = = + - + - - = + - x x x x x x x x x x x x x x b. ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) 2 2 2 5 2 4 3 5 3 2 4 4 3 4 3 5 15 2 8 3 23 4 3 4 3 3 23 4 3 x x x x x x x x x x x x x x x x x x x x x x x x x x - - + + - = - - + - + + - + + = = - + - + + = - + 6. a. ( ) ( ) ( ) 2 2 2 2 2 3 4 1 , 2 2 3 2 x x x x x x x x + + + - - ( ) ( ) 2 2 2 LCD 3 2 2 x x x = + - b. 2 2 2 2 1 3 7 , 25 4 5 x x x x x - - - + - ( )( ) ( )( ) ( )( )( ) 2 2 25 5 5 4 5 1 5 LCD 1 5 5 x x x x x x x x x x - = - + + - = - + = - - + 7. a. ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 4 4 4 4 4 2 2 2 4 2 2 2 2 2 2 4 8 2 2 8 2 2 2 2 x x x x x x x x x x x x x x x x x x x x x x x x + - + - = + - + - + - = + - + - + + + - + + = = - + - +

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