357 3.3 Zeros of Polynomial Functions 43. ƒ1x2 = 6x3 + 17x2 - 31x - 12 44. ƒ1x2 = 15x3 + 61x2 + 2x - 8 45. ƒ1x2 = 24x3 + 40x2 - 2x - 12 46. ƒ1x2 = 24x3 + 80x2 + 82x + 24 For each polynomial function, find all zeros and their multiplicities. 47. ƒ1x2 = 1x - 2231x2 - 72 48. ƒ1x2 = 1x + 1221x - 1231x2 - 102 49. ƒ1x2 = 3x1x - 221x + 321x2 - 12 50. ƒ1x2 = 5x21x2 - 1621x + 52 51. ƒ1x2 = 1x2 + x - 225Ax - 1 + 23B 2 52. ƒ1x2 = 12x2 - 7x + 323 A x - 2 - 25B Find a polynomial function ƒ1x2 of degree 3 with real coefficients that satisfies the given conditions. See Example 4. 53. Zeros of -3, 1, and 4; ƒ122 = 30 54. Zeros of 1, -1, and 0; ƒ122 = 3 55. Zeros of -2, 1, and 0; ƒ1-12 = -1 56. Zeros of 2, -3, and 5; ƒ132 = 6 57. Zero of -3 having multiplicity 3; ƒ132 = 36 58. Zero of 2 and zero of 4 having multiplicity 2; ƒ112 = -18 59. Zero of 0 and zero of 1 having multiplicity 2; ƒ122 = 10 60. Zero of -4 and zero of 0 having multiplicity 2; ƒ1-12 = -6 Find a polynomial function ƒ1x2 of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 61. 5 + i and 5 - i 62. 7 - 2i and 7 + 2i 63. 0, i, and 1 + i 64. 0, -i, and 2 + i 65. 1 + 22, 1 - 22, and 1 66. 1 - 23, 1 + 23, and 1 67. 2 - i, 3, and -1 68. 3 + 2i, -1, and 2 69. 2 and 3 + i 70. -1 and 4 - 2i 71. 1 - 22, 1 + 22, and 1 - i 72. 2 + 23, 2 - 23, and 2 + 3i 73. 2 - i and 6 - 3i 74. 5 + i and 4 - i 75. 4, 1 - 2i, and 3 + 4i 76. -1, 5 - i, and 1 + 4i 77. 1 + 2i and 2 (multiplicity 2) 78. 2 + i and -3 (multiplicity 2) Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. 79. ƒ1x2 = 2x3 - 4x2 + 2x + 7 80. ƒ1x2 = x3 + 2x2 + x - 10 81. ƒ1x2 = 4x3 - x2 + 2x - 7 82. ƒ1x2 = 3x3 + 6x2 + x + 7 83. ƒ1x2 = 5x4 + 3x2 + 2x - 9 84. ƒ1x2 = 3x4 + 2x3 - 8x2 - 10x - 1 85. ƒ1x2 = -8x4 + 3x3 - 6x2 + 5x - 7 86. ƒ1x2 = 6x4 + 2x3 + 9x2 + x + 5 87. ƒ1x2 = x5 + 3x4 - x3 + 2x + 3 88. ƒ1x2 = 2x5 - x4 + x3 - x2 + x + 5 89. ƒ1x2 = 2x5 - 7x3 + 6x + 8 90. ƒ1x2 = 11x5 - x3 + 7x - 5 91. ƒ1x2 = 5x6 - 6x5 + 7x3 - 4x2 + x + 2 92. ƒ1x2 = 9x6 - 7x4 + 8x2 + x + 6 93. ƒ1x2 = 7x5 + 6x4 + 2x3 + 9x2 + x + 5 94. ƒ1x2 = -2x5 + 10x4 - 6x3 + 8x2 - x + 1
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