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

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168 Chapter 2 Polynomial and Rational Functions 121. fx 2x 9 y 122. f x 10 Common function: y x Transformation: Vertical stretch each y-value is multiplied by 2 , then a vertical shift nine units upward −6 6 5 4 3 2 1 −3 −2 −1 1 2 x Horizontal shift of three units to the left, vertical shrink each y-value is multiplied 1 by 3, reflection in the x-axis and vertical shift of ten units upward of y x 1 3 x 3 Section 2.3 Polynomial and Synthetic Division You should know the following basic techniques and principles of polynomial division. ■ The Division Algorithm (Long Division of Polynomials) ■ Synthetic Division ■ f k is equal to the remainder of fx divided by x k (the Remainder Theorem). ■ fk 0 if and only if x k is a factor of fx. Vocabulary Check 1. f x is the dividend; dx is the divisor; gx is the quotient; rx is the remainder y 1 x2 1. and x 2 x 2 ) x 2 0x 0 x 2 2x x 2 2x 0 2x 4 x Thus, and y1 y2 . 2 4 x 2 x 2 x 2 3. y1 and (a) and (b) x5 3x3 x2 1 −9 9 6 −6 y 2 x 2 4 x 2 4 y 2 x 3 4x 4x x 2 1 −2 9 8 7 6 5 4 3 2 1 y −1 1 2 2. improper; proper 3. synthetic division 4. factor 5. remainder (c) 3 4 5 6 7 8 9 2. and 39 x Thus, and y1 y2 . 4 3x2 1 x2 x 5 2 8 39 x2 8x 5 2 8x 40 2 x 1 4 5x2 x2 5 ) x4 3x2 x 1 2 y2 x 8 2 8 39 x2 y1 5 x4 3x2 1 x2 5 x5 0x4 x3 x2 0x 1 ) x5 0x4 3x3 0x2 x 0x 0 3 4x 4x 3 0x 2 0x 4x 3 0x 2 4x 4x 0 x Thus, and y1 y2. 5 3x3 x2 1 x3 4x 4x x2 1 x
2x 4 4. and y2 x 3 x (a) and (b) 2 y1 x 1 x3 2x2 5 x2 x 1 5. 7. 9. 8 −12 12 −8 x 3 ) 2x 2 10x 12 2x 2 6x 2x 2 10x 12 x 3 4x 12 4x 12 2x 4 4x 5 ) 4x 3 7x 2 11x 5 4x 3 5x 2 4x 3 7x 2 11x 5 4x 5 x 4 2x 3 0 (c) 2x 4 6. 12x 2 11x 12x 2 15x x 4 5x 3 6x 2 x 2 x 2 x 2 3x 1 8. 4x 5 4x 5 0 x 2 3x 1 x 3 3x 2 1 10. x 2 ) x 4 5x 3 6x 2 x 2 3x 3 6x 2 3x 3 6x 2 x 2 x 2 0 x 3 3x 2 1 Section 2.3 Polynomial and Synthetic Division 169 x 2 x 1 ) x 3 2x 2 0x 5 x 3 x 2 x x 3 3x 2 x 5 3x 2 3x 3 2x 8 x Thus, and y1 y2 . 3 2x2 5 x2 2x 4 x 3 x 1 x2 x 1 x 4 ) 5x 2 17x 12 5x 2 20x 5x 2 17x 12 x 4 5x 3 3x 12 3x 12 6x 3 4x 2 0 5x 3 3x 2 ) 6x 3 16x 2 17x 6 12x 2 17x 12x 2 8x 6x 3 16x 2 17x 6 3x 2 x 3 3x 2 x 3 4x 2 3x 12 x 3 2 x 2 4x 3 9x 6 9x 6 x 2 7x 18 x 3 ) x 3 4x 2 3x 12 7x 2 3x 7x 2 21x 0 2x 2 4x 3 18x 12 18x 54 42 x 2 7x 18 42 x 3
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 and 24: 36. (a) 0 with multiplicity 2, 4 an
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: 111. 113. 115. 117. 12x 2 11x 5
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:
Page 75 and 76: 27. fx (a) Domain: all real number
Page 77 and 78: 35. gs (a) Domain: all real number
Page 79 and 80: 43. 45. f x 2x2 5x 2 2x 2 x 6
Page 81 and 82: 51. 53. 55. hx (a) Domain: all rea
Page 83 and 84: 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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