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

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146 Chapter 2 Polynomial and Rational Functions 71. Let x the first number and y the second number. 72. Let x first number and y second number. Then, Then the sum is x y S, y S x. The product is 75. (a) x y 110 ⇒ y 110 x. The product is Px xy x110 x 110x x2 . Px x 2 110x x 2 110x 3025 3025 x 55 2 3025 x 55 2 3025 The maximum value of the product occurs at the vertex of Px and is 3025. This happens when x y 55. The maximum value of the product occurs at the vertex of Px and is 72. This happens when x 12 and y 24 122 6. Thus, the numbers are 12 and 6. (c) y 1 2 x2 24x 144 144 1 2 x 122 144 1 2 x 122 72 4x 3y 200 ⇒ y 1 4 200 4x 50 x 3 3 A 2xy 2x 4 8 8x50 x 50 x x50 x 3 3 3 2000 0 0 This area is maximum when feet and y feet. 100 x 25 3 3313 —CONTINUED— x x 60 Px xy xS x. Px Sx x 2 x 2 Sx x2 Sx S2 4 S x 2 2 S2 4 S2 4 The maximum value of the product occurs at the vertex of and is S This happens when x y S2. 2 Px 4. 73. Let the first number and the second number. Then the sum is The product is Px 1 2 x2 24 x Px xy x 2 24x . x y 74. Let the first number and the second number. 24 x x 2y 24 ⇒ y . 2 Then the sum is The product is Px 1 3 x2 42 x Px xy x 3 42x . x y 42 x x 3y 42 ⇒ y . 3 1 3 x2 42x 441 441 1 3 x 212 441 1 3 x 212 147 The maximum value of the product occurs at the vertex of Px and is 147. This happens when x 21 and 42 21 y 7. Thus, the numbers are 21 and 7. 3 (b) This area is maximum when feet and y feet. 100 x A x 25 3 3313 5 600 10 1066 15 1400 20 1600 25 1666 30 1600 2 3 2 3
75. —CONTINUED— (d) A 8 x50 x 3 8 3 x2 50x 8 3 x2 50x 625 625 8 3 x 252 625 8 3 x 252 5000 3 The maximum area occurs at the vertex and is square feet. This happens when feet and feet. The dimensions are feet by 33 feet. 1 50003 x 25 y 200 4253 1003 2x 50 76. (a) Radius of semicircular ends of track: Distance around two semicircular parts of track: d 2r 2 1 r y y 2 1 2 y 77. (b) Distance traveled around track in one lap: d y 2x 200 3 y 200 2x y 4 9 x2 24 x 12 9 200 2x y Section 2.1 Quadratic Functions and Models 147 (c) Area of rectangular region: 200 2x A xy x 2 2 2 The area is maximum when and 200 250 y 100 x 50 The vertex occurs at The maximum height is y3 feet. 4 9 32 24 b 249 3. 3 12 16 2a 249 9 78. y (a) The ball height when it is punted is the y-intercept. 16 2025 x2 9 x 1.5 5 y 16 2025 02 9 0 1.5 1.5 feet 5 (b) The vertex occurs at x b 2a 95 2162025 The maximum height is —CONTINUED— f 3645 16 32 2025 3645 32 2 9 5 3645 32 1.5 6561 64 6561 32 3645 32 . 1.5 6561 64 (e) They are all identical. feet and y 33 feet 1 x 25 3 1 200x 2x2 13,122 64 x2 100x x2 100x 2500 2500 x 502 5000 96 64 . 6657 feet 104.02 feet. 64
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: 61. fx 2x 10 2 7x 30 62. x-inter
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
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Page 25 and 26: 49. 51. 53. 55. 57. 59. 61. fx x
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
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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
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Page 61 and 62: 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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79. Let fx x4 gx 5x 2. 5 10x 5
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99. fx x Rational zeros: 1 x 1 Ir
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111. 9,000,000 0.0001x 2 60x 150,
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129. 3 6i 8 3i 3 6i 8 3i 11
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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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