347 3.3 Zeros of Polynomial Functions Relating Concepts For individual or collaborative investigation (Exercises 65 –74) The remainder theorem indicates that when a polynomial ƒ1x2 is divided by x - k, the remainder is equal to ƒ1k2. Consider the polynomial function ƒ1x2 = x3 - 2x2 - x + 2. Use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of ƒ1x2. 65. ƒ 1-22 66. ƒ 1-12 67. ƒ a- 1 2b 68. ƒ102 69. ƒ112 70. ƒ a 3 2b 71. ƒ122 72. ƒ132 73. Use the results from Exercises 65–72 to plot eight points on the graph of ƒ1x2. Join these points with a smooth curve. 74. Apply the method above to graph ƒ1x2 = -x3 - x2 + 2x. Use x-values -3, -1, 1 2 , and 2 and the fact that ƒ102 = 0. 3.3 Zeros of Polynomial Functions ■ Factor Theorem ■ Rational ZerosTheorem ■ Number of Zeros ■ Conjugate ZerosTheorem ■ Zeros of a Polynomial Function ■ Descartes’ Rule of Signs Factor Theorem Consider the polynomial function ƒ1x2 = x2 + x - 2, which is written in factored form as ƒ1x2 = 1x - 121x + 22. For this function, ƒ112 = 0 and ƒ1-22 = 0, and thus 1 and -2 are zeros of ƒ1x2. Notice the special relationship between each linear factor and its corresponding zero. The factor theorem summarizes this relationship. Factor Theorem For any polynomial function ƒ1x2, x - k is a factor of the polynomial if and only if ƒ1k2 = 0. EXAMPLE 1 Determining Whether x −k Is a Factor Determine whether x - 1 is a factor of each polynomial. (a) ƒ1x2 = 2x4 + 3x2 - 5x + 7 (b) ƒ1x2 = 3x5 - 2x4 + x3 - 8x2 + 5x + 1 SOLUTION (a) By the factor theorem, x - 1 will be a factor if ƒ112 = 0. Use synthetic division and the remainder theorem to decide. 1)2 0 3 -5 7 2 2 5 0 2 2 5 0 7 ƒ112 = 7 The remainder is 7, not 0, so x - 1 is not a factor of 2x4 + 3x2 - 5x + 7. Use a zero coefficient for the missing term.
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