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SOLUTIONS MANUAL

ALGEBRA AND TRIGONOMETRY TWELFTH EDITION

Michael Sullivan

Table of Contents Chapter R Review R.1 Real Numbers ............................................................................................................................ 1 R.2 Algebra Essentials ..................................................................................................................... 5 R.3 Geometry Essentials ................................................................................................................ 11 R.4 Polynomials ............................................................................................................................. 16 R.5 Factoring Polynomials ............................................................................................................ 23 R.6 Synthetic Division ................................................................................................................... 28 R.7 Rational Expressions ............................................................................................................... 30 R.8 nth Roots; Rational Exponents ................................................................................................ 40

Chapter 1 Equations and Inequalities 1.1 Linear Equations ...................................................................................................................... 50 1.2 Quadratic Equations................................................................................................................. 68 1.3 Complex Numbers; Quadratic Equations in the Complex Number System............................ 86 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations................................ 92 1.5 Solving Inequalities ............................................................................................................... 116 1.6 Equations and Inequalities Involving Absolute Value .......................................................... 127 1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications ...... 137 Chapter Review ............................................................................................................................ 145 Chapter Test .................................................................................................................................. 153 Chapter Projects ............................................................................................................................ 155

Chapter 2 Graphs 2.1 The Distance and Midpoint Formulas ................................................................................... 156 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ............................................. 169 2.3 Lines ...................................................................................................................................... 183 2.4 Circles .................................................................................................................................... 201 2.5 Variation ................................................................................................................................ 215 Chapter Review ............................................................................................................................ 221 Chapter Test .................................................................................................................................. 227 Cumulative Review ...................................................................................................................... 229 Chapter Project ............................................................................................................................. 231

Chapter 3 Functions and Their Graphs 3.1 Functions................................................................................................................................ 232 3.2 The Graph of a Function ........................................................................................................ 250 3.3 Properties of Functions .......................................................................................................... 259 3.4 Library of Functions; Piecewise-defined Functions .............................................................. 276 3.5 Graphing Techniques: Transformations ................................................................................ 288 3.6 Mathematical Models: Building Functions ........................................................................... 306 Chapter Review ............................................................................................................................ 314 Chapter Test .................................................................................................................................. 321 Cumulative Review ...................................................................................................................... 324 Chapter Projects ............................................................................................................................ 328

Chapter 11 Analytic Geometry 11.2 The Parabola ...................................................................................................................... 1199 11.3 The Ellipse ......................................................................................................................... 1215 11.4 The Hyperbola ................................................................................................................... 1232 11.5 Rotation of Axes; General Form of a Conic ...................................................................... 1252 11.6 Polar Equations of Conics ................................................................................................. 1265 11.7 Plane Curves and Parametric Equations ............................................................................ 1274 Chapter Review .......................................................................................................................... 1289 Chapter Test ................................................................................................................................ 1298 Cumulative Review .................................................................................................................... 1303 Chapter Projects .......................................................................................................................... 1305

Chapter 12 Systems of Equations and Inequalities 12.1 Systems of Linear Equations: Substitution and Elimination ............................................. 1309 12.2 Systems of Linear Equations: Matrices ............................................................................. 1332 12.3 Systems of Linear Equations: Determinants...................................................................... 1356 12.4 Matrix Algebra................................................................................................................... 1370 12.5 Partial Fraction Decomposition ......................................................................................... 1389 12.6 Systems of Nonlinear Equations ........................................................................................ 1407 12.7 Systems of Inequalities ...................................................................................................... 1435 12.8 Linear Programming .......................................................................................................... 1450 Chapter Review .......................................................................................................................... 1464 Chapter Test ................................................................................................................................ 1479 Cumulative Review .................................................................................................................... 1487 Chapter Projects .......................................................................................................................... 1491

Chapter 13 Sequences; Induction; the Binomial Theorem 13.1 Sequences .......................................................................................................................... 1493 13.2 Arithmetic Sequences ........................................................................................................ 1503 13.3 Geometric Sequences; Geometric Series ........................................................................... 1512 13.4 Mathematical Induction ..................................................................................................... 1524 13.5 The Binomial Theorem ...................................................................................................... 1533 Chapter Review .......................................................................................................................... 1540 Chapter Test ................................................................................................................................ 1544 Cumulative Review .................................................................................................................... 1547 Chapter Projects .......................................................................................................................... 1550

Chapter 14 Counting and Probability 14.1 Counting ............................................................................................................................ 1553 14.2 Permutations and Combinations ........................................................................................ 1556 14.3 Probability.......................................................................................................................... 1561 Chapter Review .......................................................................................................................... 1568 Chapter Test ................................................................................................................................ 1570 Cumulative Review .................................................................................................................... 1571 Chapter Projects .......................................................................................................................... 1574

Appendix Graphing Utilities Section 1 The Viewing Rectangle ............................................................................................. 1577 Section 2 Using a Graphing Utility to Graph Equations ........................................................... 1578 Section 3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ................. 1583 Section 5 Square Screens ........................................................................................................... 1585

Chapter R Review Section R.1

16.

( A  B)  C  1, 3, 4,5, 9  2, 4, 6, 7,8   1,3, 4, 6

1. rational

 4  1, 3, 4, 6

2. 4  5  6  3  4  30  3  31

 1,3, 4, 6

3. Distributive

17. A  0, 2, 6, 7, 8

4. c 5. a

18. C  0, 2, 5, 7, 8, 9

6. b

19. A  B  1, 3, 4, 5, 9  2, 4, 6, 7, 8  4  0, 1, 2, 3, 5, 6, 7, 8, 9

7. True 8. False; The Zero-Product Property states that if a product equals 0, then at least one of the factors must equal 0.

20. B  C  2, 4, 6, 7, 8  1, 3, 4, 6  1, 2, 3, 4, 6, 7, 8  0, 5, 9

9. False; 6 is the Greatest Common Factor of 12 and 18. The Least Common Multiple is the smallest value that both numbers will divide evenly. The LCM for 12 and 18 is 36.

21. A  B  0, 2, 6, 7, 8  0, 1, 3, 5, 9  0, 1, 2, 3, 5, 6, 7, 8, 9

22. B  C  0, 1, 3, 5, 9  0, 2, 5, 7, 8, 9

10. True

 0, 5, 9

11. A  B  1, 3, 4,5, 9  2, 4, 6, 7,8

 1, 2,3, 4, 5, 6, 7,8, 9

23. a.

12. A  C  1, 3, 4,5, 9  1, 3, 4, 6  1, 3, 4, 5, 6, 9

13. A  B  1, 3, 4,5, 9  2, 4, 6, 7,8  4

6, 2,5



 



14. A  C  1, 3, 4,5, 9  1, 3, 4, 6  1, 3, 4 15.

( A  B)  C  1, 3, 4,5, 9  2, 4, 6, 7,8   1,3, 4, 6

24. a.

 1, 2,3, 4,5, 6, 7,8,9  1,3, 4, 6  1, 3, 4, 6

2,5



1 6, , 1.333..., 2,5 2



1 6, , 1.333...,  , 2,5 2

1

0,1



 5



5  , 2.060606...  2.06,1.25, 0,1 3

Chapter R: Review

e. 25. a.



5  , 2.060606...  2.06,1.25, 0,1, 5 3

1

0,1



1 1 1 0,1, , , 2 3 4

d. None e.





1 1 1 0,1, , , 2 3 4

 

None

1

1.3, 1.2, 1.1, 1

d. e. 28. a.

9.999

9.998

36. a.

1.001

1.000

37. a.

0.429

0.428

38. a.

0.556

0.555

39. a.

34.733

40. a.

16.200

45. 3 y  1  2 46. 2 x  4  6

None

47. x  2  6

None

  

48. 2  y  6

  

1 2,  , 2  1,   2 2, , 2  1, 

1 2

None



35. a.

44. 3  y  2  2

1.3, 1.2, 1.1, 1

b. None

0.053

43. x  2  3  4

b. None c.

42. 5  2  10

d. None

27. a.

0.054

41. 3  2  5

26. a.

34. a.

49.

x 6 2

50.

2 6 x

51. 9  4  2  5  2  7



52. 6  4  3  2  3  5

1  10.3 2

 2,   2



53. 6  4  3  6  12  6 54. 8  4  2  8  8  0



1  2,   2,  10.3 2

55. 18  5  2  18  10  8

29. a.

18.953

18.952

30. a.

25.861

31. a.

28.653

32. a.

99.052

33. a.

0.063

0.062

56. 100  10  2  100  20  80 57. 4 

1 12  1 13   3 3 3

58. 2 

1 4 1 3   2 2 2

Section R.1: Real Numbers 59. 6  3  5  2   3  2    6  15  2  1   6  17  11 60. 2  8  3   4  2    3  2  8  3   6    3  2  8  18  3  2   10  3

70.

5 3 53 53 1     9 10 3  3  5  2 3  3  5  2 6

71.

6 10 2  3  5  2 2  3  5  2 4     25 27 5  5  3  9 5  5  3  9 45

72.

21 100 3  7  4  25 3  7  4  25  28    25 3 25  3 25  3

73.

3 2 15  8 23    4 5 20 20

74.

4 1 8  3 11    3 2 6 6

75.

7 4 49  32 81    8 7 56 56

76.

8 15 16  135 151    9 2 18 18

77.

5 1 10  3 13    18 12 36 36

78.

2 8 6  40 46    15 9 45 45

79.

5 8 25  64 39 13     24 15 120 120 40

80.

3 2 94 5    14 21 42 42

81.

3 2 98 1    20 15 60 60

82.

6 3 12  15 3    35 14 70 70

 20  3  23

61. 4   9  5   6  7  3  4  14   42  3  56  42  3  14  3  17

62. 1   4  3  2  2   1  12  2  2   1  12  11

63. 10  6  2  2   8  3   2  10   6  4  5  2  10   2  5  2  10   7   2  10  14  4

64. 2  5  4   6   3  4    2  20  6   1   18   6  18  6  12

65.

 5  3 2   2  2  1

66.

 5  4  13   9  13  3

67.

4  8 12  6 53 2

68.

2  4 2   1 53 2

69.

3 10 3  2  5 3  2  5 2     5 21 5  3  7 5  3  7 7

5  18  5 27 5  9  3 5  9  3 15   83.       11  18 11 9  2 11 9  2 11 22  27     5  21  5 35 5  7  5 5  7  5 25 84.         2  21 2 7  3  2 7  3  2 6  35   

Chapter R: Review

85.

86.

1 4 17 4 17 21      1 3 7 21 21 21 21

98.

 3x  1 x  5   3x 2  15 x  x  5  3 x 2  14 x  5

2 4 1 2 22 2 2 2 2 2         3 5 6 3 5  3  2 3 5  3  2 3 15 2 5 2 10 2 10  2 12        3 5 15 15 15 15 15 43 4 3 4    53 5 3 5

99.

 x  8  x  2   x 2  2 x  8 x  16  x 2  10 x  16

100.

 x  4  x  2   x 2  2 x  4 x  8  x2  6 x  8

101. 3x( x  5k )  3x 2  60 x

3 3 2 3 3 6 3 6 2 3 87. 2           4 8 1 4 8 4 8 4 2 8 12 3 12  3 15     8 8 8 8

3x 2  15 xk  3x 2  60 x 15 xk  60 x k 4

5 1 3 5 1 35 1 3 5 1 88. 3          6 2 1 6 2 3 2 2 3  2 2 5 1 5 1 4     2 2 2 2 2

102.

( x  k )( x  3k )  x 2  4 x  12 x 2  3kx  kx  3k 2  x 2  4 x  12 x 2  x(3k  k )  3k 2  x 2  4 x  12 x 2  x(3k  k )  3k 2  x 2  4 x  12

89. 6  x  4   6 x  24

x 2  x(2k )  3k 2  x 2  4 x  12 2k  4

90. 4  2 x  1  8 x  4

k 2

91. x  x  4   x  4 x 2

103. 2 x  3 x  2  x  3  x   2  3  x

92. 4 x  x  3  4 x 2  12 x

  5  x  5x

1 3 1 2  3x 2 3  93. 2  x    2  x  2   4 2 4 2 22 2   2  3x 2 3    x 1 2 2 2 2

104. 2  3  4  2  12  14 since multiplication comes before addition in the order of operations for real numbers.

 2  3  4  5  4  20

1 2 1 3 2x 3 2  94. 3  x    3  x  3   3 6 3 6 3 3 2   3  2x 3 1    2x  3 3 2 2

95.

since operations inside parentheses come before multiplication in the order of operations for real numbers. 105. 2  3  4   2 12   24

 x  2  x  4   x 2  4 x  2 x  8

 2  3   2  4    6 8   48

 x  6x  8

96.

106.

 x  5 x  1  x 2  x  5 x  5  x2  6 x  5

97.

 x  9  2 x  7   2 x 2  7 x  18 x  63

43 7   1 , but 25 7 4 3 4  5  3  2 20  6 26 13       2.6 2 5 10 10 10 5

107. Subtraction is not commutative; for example: 2  3  1  1  3  2 .

 2 x 2  11x  63