C H A P T E R 2 Polynomial and Rational Functions

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248 Chapter 2 Polynomial and Rational Functions 91. 93. 95. 96. f x x 3 2x 2 21x 18 92. Possible rational zeros: ±1, ±2, ±3, ±6, ±9, ±18 1 1 1 2 1 3 21 3 18 18 18 0 x 3 2x 2 21x 18 x 1x 2 3x 18 x 1x 6x 3 The zeros of f x are x 1, x 6, and x 3. f x x 3 10x 2 17x 8 94. Possible rational zeros: ±1, ±2, ±4, ±8 1 1 1 10 1 9 17 9 8 8 8 0 x 3 10x 2 17x 8 x 1x 2 9x 8 x 1x 1x 8 x 1 2 x 8 The zeros of f x are x 1 and x 8. f x x 4 x 3 11x 2 x 12 Possible rational zeros: ±1, ±2, ±3, ±4, ±6, ±12 3 1 1 4 1 1 1 3 4 4 4 0 11 12 1 1 0 1 4 4 0 x 4 x 3 11x 2 x 12 x 3x 4x 2 1 1 3 4 12 12 0 The real zeros of f x are x 3, and x 4. f x 25x 4 25x 3 154x 2 4x 24 3 25 25 2 25 25 75 50 50 50 So, f x 25x 4 25x3 154x2 25 0 4 0 4x 24 x 3x 225x 2 4 x 3x 25x 25x 2. Zeros: x 3, 2, ± 2 5 154 150 4 4 0 8 8 4 12 8 24 24 0 f x 3x 3 20x 2 7x 30 Possible rational zeros: 1 3 3 20 3 23 So, f x 3x 3 20x 2 7x 30 x 13x 2 23x 30 x 13x 5x 6 0 x 13x 5x 6. Zeros: x 1, 5 3 , 6 7 23 30 ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30, ± 1 2 5 3 , ± 3 , ± 3 , 30 30 0 f x x 3 9x 2 24x 20 ± 10 3 Possible rational zeros: ±1, ±2, ±4, ±5, ±10, ±20 5 1 9 5 So, f x x3 9x2 1 4 4 0 24x 20 x 5x 2 4x 4 x 5x 2 2 . Zeros: x 5, 2 Possible rational zeros: ± 6 25, ± 8 25, ± 12 25, ± 24 ± 25 4 ± 25, 1 25, ± 2 25, ± 3 ± 25, 24 ±24, ± 5 , 1 5, ± 2 5, ± 3 5, ± 4 5, ± 6 5, ± 8 5, ± 12 ±1, ±2, ±3, ±4, ±6, ±8, ±12, 5 , 24 20 20 20
97. Since is a zero, so is 3x 2x 4x Multiply by 3 to clear the fraction. 2 fx 3x 3i 3i. 3 2 3x 4x 3ix 3i 3x 2 14x 8x 2 3 3x 4 14x 3 17x 2 42x 24 Review Exercises for Chapter 2 249 Note: where a is any real nonzero number, has zeros 2 , 4, and ±3i. f x a3x4 14x3 17x2 42x 24, 98. Since is a zero and the coefficients are real, must also be a zero. x2 x 6x 12 1 2i 99. fx x Zero: i 1 2i Since i is a zero, so is i. f x x 2x 3x 1 2ix 1 2i i 1 4 1 4 4 i 1 4i 4 3 4x2 x 4, x 2 x 6x 2 2x 5 x 4 x 3 3x 2 17x 30 4i 1 4i 1 1 2 4i 4i 1 2 0 hx x 4ix 4ix 2 Zeros: x ± 4i, 2 2 4i 2 4i 16 16 8i 8i 8i 8i 32 32 0 1 i 1 1 4 i 4 i i 4 4i 4i 4i fx x ix ix 4, Zeros: x ± i, 4 100. hx x Since 4i is a zero, so is 4i. 3 2x2 16x 32 101. gx 2x Zero: 2 i Since 2 i is a zero, so is 2 i 4 3x3 13x2 37x 15, 102. 2 i 2 2 2 i 2 3 4 2i 1 2i 1 2i 4 2i gx x 2 ix 2 i2x2 2 5 3 0 5x 3 Zeros: x 2 ± i, 1 2 , 3 fx 4x4 11x3 14x2 6x 103. fx x3 4x2 5x x4x 3 11x 2 14x 6 One zero is x 0. Since 1 i is a zero, so is 1 i. 1 i 4 4 1 i 4 4 11 4 4i 7 4i 7 4i 4 4i 3 f x xx 1 ix 1 i4x 3 xx 1 ix 1 i4x 3 Zeros: 0, 3 4 , 1 i, 1 i 14 11 3i 3 3i 3 3i 3 3i 0 6 6 0 0 x 2 ix 2 i2x 1x 3 xx 2 4x 5 xx 5x 1 Zeros: x 0, 5, 1 3 13 5i 13 5i 0 13 5i 10 5i 37 31 3i 6 3i 6 3i 6 3i 15 15 0
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CHAPTER 2 Polynomial and Rational F
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9. (a) (c) 10. (a) (c) y 1 2 x2 (b
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13. f x x 2 5 14. Vertex: Axis of
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25. 27. 29. hx 4x 2 4x 21 Vertex
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41. 2, 2 is the vertex. 42. 2, 0 is
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61. fx 2x 10 2 7x 30 62. x-inter
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75. —CONTINUED— (d) A 8 x50 x
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86. (a) and (c) (b) (d) 1996 (e) Ve
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Section 2.2 Polynomial Functions of
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11. (a) (b) fx x4 fx x 3 3 4 y
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Section 2.2 Polynomial Functions of
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36. (a) 0 with multiplicity 2, 4 an
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49. 51. 53. 55. 57. 59. 61. fx x
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70. 72. 74. gx x 2 10x 16 x 2x
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81. 83. 85. Zeros: 0, 2, 2 all of m
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92. (a) 93. (c) 200 7 140 V 4 3r 3
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111. 113. 115. 117. 12x 2 11x 5
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2x 4 4. and y2 x 3 x (a) and (b
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20. 22. 24. 26. 28. 29. 31. 33. 3 5
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45. fx 4x (a) 1 4 0 13 4 4 4 4 9 3
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58. f x 3x Factors: x 3, x 2 3
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67. 69. 71. ht t 3 2t 2 7t 2 (a
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79. 83. x n 3 ) x 3n 9x 2n 27x n
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1. 4. 7. a 10 b 6 Section 2.4 Comp
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52. 53. 55. 57. 59. 8 16i 2i 3i 4
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74. 4.5x2 3x 12 0; a 4.5, b 3,
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Section 2.5 Zeros of Polynomial Fun
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18. 19. f x 3x 3 19x 2 33x 9 Po
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−24 (c) Real zeros: 2, 1 145 ± 8
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47. 48. fx 2x 3 3x 2 50x 75 Sin
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54. f x x Since 1 3i is a zero, s
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Page 79 and 80: 43. 45. f x 2x2 5x 2 2x 2 x 6
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Page 83 and 84: 60. gx (a) Domain: all real number
Page 85 and 86: 72. (a) x-intercepts: (b) 0 x 3
Page 87 and 88: 80. (a) 81. False. Polynomial funct
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Page 95 and 96: 39. −2 −1 0 1 2 3 4 5 x Section
Page 97 and 98: Section 2.7 Nonlinear Inequalities
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Page 101 and 102: 73. C 0.0031t 3 0.216t 2 5.54t
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Page 127 and 128: Problem Solving for Chapter 2 261 1
Page 129: 19. Use synthetic division to show
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C H A P T E R 2 Polynomial and Rational Functions

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