SECTION 4.1 Polynomial Functions 203 In Problems 51–60, find a polynomial function with the given real zeros whose graph contains the given point. 59. Zeros: 5− (multiplicity 2), 2 (multiplicity 1), 4 (multiplicity 1); degree 4; contains the point 3, 128 ( ) 60. Zeros: 4− (multiplicity 1), 0 (multiplicity 3), 2 (multiplicity 1); degree 5; contains the point 2, 64 ( ) − 57. Zeros: 1− (multiplicity 2), 1 (multiplicity 2) Degree 4 Point: 2, 45 ( ) − 58. Zeros: 0 (multiplicity 1), 1− (multiplicity 2), 3 (multiplicity 2) Degree 5 Point: 1, 48 ( ) − In Problems 61–72, for each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of x . 61. f x x x 3 7 3 2 ( ) ( )( ) = − + 62. f x x x 4 4 3 3 ( ) ( )( ) = + + 63. f x x x 7 4 5 2 2 3 ( ) ( ) ( ) = + − 64. f x x x 2 3 4 2 3 ( ) ( ) ( ) = − + 65. f x x x 2 1 2 4 2 3 ( ) ( ) ( ) = − + + 66. f x x x 1 3 1 2 3 ( ) ( ) ( ) = − − 67. f x x x 5 4 3 2 ( ) ( ) ( ) = − + 68. f x x x 3 2 2 4 ( ) ( ) ( ) = + − 69. f x x x 1 2 2 9 7 2 2 2 ( ) ( ) ( ) = + + 70. f x x 2 3 2 3 ( ) ( ) = − + 71. f x x x 2 2 2 2 ( ) ( ) = − − 72. f x x x 4 3 2 ( ) ( ) = − In Problems 73–76, identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, state why not. 73. x y –4 –2 2 4 4 2 –4 –2 74. x y –4 –2 2 4 4 2 –4 –2 75. x y –2 2 2 –2 76. x y 22 24 2 4 22 4 2 In Problems 77–80, find a polynomial function that might have the given graph. (More than one answer may be possible.) 77. x y 0 1 2 78. x y 0 1 2 79. x y 2 3 2 1 –1 –2 1 –2 –1 80. x y 2 3 2 1 –1 –2 1 –2 –1 51. Zeros: 2, 3, 5 − Degree 3 Point: 2, 36 ( ) 52. Zeros: 2, 0, 2 − Degree 3 Point: 4, 16 ( ) − 53. Zeros: 2, 0, 1, 3 − Degree 4 Point: 1 2 , 63 ( ) − − 54. Zeros: 5, 1, − − 2, 6 Degree 4 Point: 5 2 , 15 ( ) 55. Zeros: 3, 1, 4 − Degree 3 y-intercept: 36 56. Zeros: 4, 1, 2 − − Degree 3 y-intercept: 16
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