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

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158 Chapter 2 Polynomial and Rational Functions 42. (a) 44. x ±5, 4, all with multiplicity 1 (b) The multiplicity of x 5 is 1. (c) f x x 3 4x 2 25x 100 43. The multiplicity of x 5 is 1. The multiplicity of x 4 is 1. There are two turning points. −9 0 x 2 x 4 25x 4 0 x 2 25x 4 0 x 5x 5x 4 140 −20 9 y 4x 3 4x 2 8x 8 45. (a) (b) (c) −3 −11 (1, 0, 1.414214, 0, 1.414214, 0 0 4x 3 4x 2 8x 8 0 4x 2 x 1 8x 1 0 4x 2 8x 1 0 4x 2 2x 1 x ±2, 1 (d) The intercepts match part (b). 46. y (a) 12 1 4 x 3x 2 9 47. 2 −18 18 −12 (b) 0, 0, 3, 0, 3, 0 fx x 2 10x Note: fx ax 0x 10 axx 10 has zeros 0 and 10 for all real numbers a 0. 3 (c) y 4x 3 20x 2 25x (a) (b) x-intercepts: (c) −2 12 −4 0 4x 3 20x 2 25x 0 x2x 5 2 x 0 or x 5 2 0, 0, 5 2 , 0 (d) The solutions are the same as the x-coordinates of the x-intercepts. y x 5 5x 3 4x (a) (b) x-intercepts: 0, 0, ±1, 0, ±2, 0 (c) 0 x 0, ±3 1 4x3x 2 9 −6 6 −4 0 x 5 5x 3 4x 0 xx 2 1x 2 4 4 0 xx 1x 1x 2x 2 x 0, ±1, ±2 x-intercepts: 0, 0, ±3, 0 (d) The intercepts match part (b). (d) The solutions are the same as the x-coordinates of the x-intercepts. fx x 0x 10 48. f x x 0x 3 xx 3 x 2 3x 6 Note: f x axx 3 has zeros 0 and 3 for all real numbers a.
49. 51. 53. 55. 57. 59. 61. fx x 2x 6 50. fx x 2x 6 fx x 2 4x 12 Note: fx ax 2x 6 has zeros 2 and 6 for all real numbers a 0. fx x 0x 2x 3 52. xx 2x 3 x 3 5x 2 6x Note: fx axx 2x 3 has zeros 0, 2, 3 for all real numbers a 0. fx x 4x 3x 3x 0 54. x 4x 2 9x x 4 4x 3 9x 2 36x Note: fx ax has these zeros for all real numbers a 0. 4 4x3 9x2 36x fx x 1 3 x 1 3 x 1 3 x 1 3 x 1 2 3 2 x 2 2x 1 3 x 2 2x 2 Note: fx ax has these zeros for all real numbers a 0. 2 2x 2 fx x 2x 2 x 2 2 x 2 4x 4 Note: f x ax a 0, has degree 2 and zero x 2. 2 4x 4, f x x 3x 0x 1 xx 3x 1 x 3 2x 2 3x Note: f x ax a 0, has degree 3 and zeros x 3, 0, 1. 3 2x2 3x, xx 3x 3 x 3 3x Note: f x ax a 0, has degree 3 and zeros x 0, 3, 3. 3 3x, Section 2.2 Polynomial Functions of Higher Degree 159 56. 58. 60. f x x 4x 5 x 4x 5 x 2 x 20 Note: f x ax 4x 5 has zeros 4 and 5 for all real numbers a. f x x 0x 2x 5 xx 2x 5 xx 2 7x 10 x 3 7x 2 10x Note: f x axx 2x 5 has zeros 0, 2, 5 for all real numbers a. f x x 2x 1x 0x 1x 2 xx 2x 1x 1x 2 xx 2 4x 2 1 xx 4 5x 2 4 x 5 5x 3 4x Note: f x axx 2x 1x 1x 2 has zeros 2, 1, 0, 1, 2 for all real numbers a. x 2x 42 f x x 2x 4 5 x 4 5 x 2x 4 5x 4 5 5 xx 4 2 5x 2x 4 2 10 x 3 8x 2 16x 5x 2x 2 16x 32 10 x 3 10x 2 27x 22 Note: f x ax has these zeros for all real numbers a. 3 10x 2 27x 22 f x x 8x 4 x 8x 4 x 2 12x 32 Note: f x ax has degree 2 and zeros x 8 and 4. 2 12x 32, a 0, f x x 2x 4x 7 x 2x 2 11x 28 x 3 9x 2 6x 56 Note: f x ax has degree 3 and zeros x 2, 4, and 7. 3 9x2 6x 56, a 0, f x x 0x 3x 3 62. f x x 93 x3 27x2 243x 729 Note: f x ax has degree 3 and zero x 9. 3 27x2 243x 729, a 0,
Page 1 and 2: CHAPTER 2 Polynomial and Rational F
Page 3 and 4: 9. (a) (c) 10. (a) (c) y 1 2 x2 (b
Page 5 and 6: 13. f x x 2 5 14. Vertex: Axis of
Page 7 and 8: 25. 27. 29. hx 4x 2 4x 21 Vertex
Page 9 and 10: 41. 2, 2 is the vertex. 42. 2, 0 is
Page 11 and 12: 61. fx 2x 10 2 7x 30 62. x-inter
Page 13 and 14: 75. —CONTINUED— (d) A 8 x50 x
Page 15 and 16: 86. (a) and (c) (b) (d) 1996 (e) Ve
Page 17 and 18: Section 2.2 Polynomial Functions of
Page 19 and 20: 11. (a) (b) fx x4 fx x 3 3 4 y
Page 21 and 22: Section 2.2 Polynomial Functions of
Page 23: 36. (a) 0 with multiplicity 2, 4 an
Page 27 and 28: 70. 72. 74. gx x 2 10x 16 x 2x
Page 29 and 30: 81. 83. 85. Zeros: 0, 2, 2 all of m
Page 31 and 32: 92. (a) 93. (c) 200 7 140 V 4 3r 3
Page 33 and 34: 111. 113. 115. 117. 12x 2 11x 5
Page 35 and 36: 2x 4 4. and y2 x 3 x (a) and (b
Page 37 and 38: 20. 22. 24. 26. 28. 29. 31. 33. 3 5
Page 39 and 40: 45. fx 4x (a) 1 4 0 13 4 4 4 4 9 3
Page 41 and 42: 58. f x 3x Factors: x 3, x 2 3
Page 43 and 44: 67. 69. 71. ht t 3 2t 2 7t 2 (a
Page 45 and 46: 79. 83. x n 3 ) x 3n 9x 2n 27x n
Page 47 and 48: 1. 4. 7. a 10 b 6 Section 2.4 Comp
Page 49 and 50: 52. 53. 55. 57. 59. 8 16i 2i 3i 4
Page 51 and 52: 74. 4.5x2 3x 12 0; a 4.5, b 3,
Page 53 and 54: Section 2.5 Zeros of Polynomial Fun
Page 55 and 56: 18. 19. f x 3x 3 19x 2 33x 9 Po
Page 57 and 58: −24 (c) Real zeros: 2, 1 145 ± 8
Page 59 and 60: 47. 48. fx 2x 3 3x 2 50x 75 Sin
Page 61 and 62: 54. f x x Since 1 3i is a zero, s
Page 63 and 64: 69. 71. 73. gx x 4 4x 3 8x 2 16
Page 65 and 66: 79. Let fx x4 gx 5x 2. 5 10x 5
Page 67 and 68: 99. fx x Rational zeros: 1 x 1 Ir
Page 69 and 70: 111. 9,000,000 0.0001x 2 60x 150,
Page 71 and 72: 129. 3 6i 8 3i 3 6i 8 3i 11
Page 73 and 74: 2 x x 2 7. fx 2 x x 2 Domain:
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27. fx (a) Domain: all real number
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35. gs (a) Domain: all real number
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43. 45. f x 2x2 5x 2 2x 2 x 6
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51. 53. 55. hx (a) Domain: all rea
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60. gx (a) Domain: all real number
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72. (a) x-intercepts: (b) 0 x 3
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80. (a) 81. False. Polynomial funct
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4. 5. 8. 10. 3x2 x2 < 1 4 (a) x 2
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17. 19. 20. x 2 2x 3 < 0 18. x 3
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25. 27. 29. 31. 4x 2 4x 1 ≤ 0 2
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39. −2 −1 0 1 2 3 4 5 x Section
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Section 2.7 Nonlinear Inequalities
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59. 61. x x x ≥ 0 x 5x 7 Critic
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73. C 0.0031t 3 0.216t 2 5.54t
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Review Exercises for Chapter 2 1. (
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9. 11. 13. hx 4x2 4x 13 10. f x
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21. C 70,000 120x 0.055x The min
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41. fx xx 3 x 2 5x 3 (a) The de
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58. f x 3x (a) 4 3 8 20 16 12 16 1
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77. 79. 4 2 4 2 3i 2 1 i 2
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97. Since is a zero, so is 3x 2x
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111. 114. 117. f x Domain: all rea
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4 126. hx x 1 (a) Domain: all rea
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133. 135. fx 3x3 2x 2 3x 2 3x 2
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142. 144. 146. 148. 12x 3 20x 2 <
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3. V l w h x 2 x 3 x 2 x 3 2
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Problem Solving for Chapter 2 261 1
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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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