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

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250 Chapter 2 Polynomial and Rational Functions 104. gx x 3 7x 2 36 2 1 1 7 2 9 0 18 18 36 36 0 The zeros of x are x 3, 6. The zeros of gx are 2, 3, 6. gx x 2x 3x 6 2 9x 18 x 3x 6 105. gx x Zero: x 4 4 4x3 3x2 40x 208, 106. 4 1 1 4 1 1 4 4 0 0 4 4 3 0 3 3 16 13 40 12 52 52 52 gx x 4 2 x 2 4x 13 By the Quadratic Formula the zeros of x are x 2 ± 3i. The zeros of gx are x 4 of multiplicity 2, and x 2 ± 3i. 2 4x 13 gx x 4 2 x 2 3ix 2 3i x 4 2 x 2 3ix 2 3i 0 208 208 107. gx 5x gx has two variations in sign, so g has either two or no positive real zeros. 3 3x2 6x 9 gx 5x 3 3x 2 6x 9 gx has one variation in sign, so g has one negative real zero. 4 3 1 4 4 1 5 3 5 4 17 4 Since the last row entries alternate in sign, x is a lower bound. 1 4 0 fx x 4 8x 3 8x 2 72x 153 3 1 1 3 1 1 8 3 11 11 3 8 8 33 41 41 24 72 123 17 51 51 51 By the Quadratic Formula, the zeros of x are 2 8x 17 x 8 ± 82 4117 21 153 153 The zeros of fx are 3, 3, 4 i, 4 i. fx x 3x 3x 4 ix 4 i 0 8 ± 4 2 4 ± i. 108. hx 2x hx has three variations in sign, so h has either three or one positive real zeros. 5 4x3 2x2 5 hx 2x 5 4x 3 2x 2 5 2x 5 4x 3 2x 2 5 hx has two variations in sign, so h has either two or no negative real zeros. 109. (a) Since the last row has all positive entries, is an upper bound. (b) 1 fx 4x 1 4 3 4 3 4 1 5 4 1 5 2 x 1 4 4 3 3x2 4x 3 110. gx 2x (a) 8 2 5 14 8 16 88 592 2 11 74 600 Since the last row has all positive entries, x 8 is an upper bound. (b) 4 2 5 8 14 52 8 152 3 5x2 14x 8 2 13 38 144 Since the last row entries alternate in sign, x 4 is a lower bound.
111. 114. 117. f x Domain: all real numbers except x 12 5x x 12 f x x2 x 2 x 2 4 Domain: all real numbers hx 2x 10 x 2 2x 15 2x 5 x 3x 5 2 , x 5 x 3 Vertical asymptote: x 3 Horizontal asymptote: y 0 x 112. 119. fx (a) Domain: all real numbers x except x 0 5 x2 (b) No intercepts (c) Vertical asymptote: x 0 Horizontal asymptote: y 0 (d) x ±3 5 9 1 −1 1 2 −2 −3 y ±2 y 5 5 4 ±1 x f x 3x2 1 3x 1 3x 0 3x 1 x 1 3 Domain: all real numbers except x 1 3 115. fx Vertical asymptote: x 3 4 x 3 Horizontal asymptote: y 0 x Review Exercises for Chapter 2 251 113. f x 8 x 2 10x 24 8 x 4x 6 Domain: all real numbers except x 4 and x 6 116. fx Vertical asymptote: none 2x2 5x 3 x2 2 Horizontal asymptote: y 2 118. hx Vertical asymptotes: x 2, x 1 x3 4x2 x2 3x 2 x2x 4 x 2x 1 Horizontal asymptotes: none 120. fx (a) Domain: all real numbers x except x 0 4 x (b) No intercepts (c) Vertical asymptote: x 0 Horizontal asymptote: y 0 (d) x y 3 4 3 4 3 2 1 −3 −2 −1 1 2 3 4 −2 −3 y 2 2 1 4 x 1 2 3 4 2 4 3 x
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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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69. 71. 73. gx x 4 4x 3 8x 2 16
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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
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Page 95 and 96: 39. −2 −1 0 1 2 3 4 5 x Section
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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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