344 CHAPTER 3 Polynomial and Rational Functions −8 −5 8 5 Figure 16 −8 −4 8 4 Figure 17 Potential Zeros of Polynomial Functions A zero of a polynomial function ƒ1x2 is a number k such that ƒ1k2 = 0. Real number zeros are the x-values of the x-intercepts of the graph of the function. The remainder theorem gives a quick way to determine whether a number k is a zero of a polynomial function ƒ1x2, as follows. 1. Use synthetic division to find ƒ1k2. 2. If the remainder is 0, then ƒ1k2 = 0 and k is a zero of ƒ1x2. If the remainder is not 0, then k is not a zero of ƒ1x2. A zero of ƒ1x2 is a root, or solution, of the equation ƒ1x2 = 0. EXAMPLE 3 Determining Whether a Number Is a Zero Determine whether the given number k is a zero of ƒ1x2. (a) ƒ1x2 = x3 - 4x2 + 9x - 6; k = 1 (b) ƒ1x2 = x4 + x2 - 3x + 1; k = -1 (c) ƒ1x2 = x4 - 2x3 + 4x2 + 2x - 5; k = 1 + 2i SOLUTION (a) To determine whether 1 is a zero of ƒ1x2 = x3 - 4x2 + 9x - 6, use synthetic division. Proposed zero 1)1 -4 9 -6 ƒ1x2 = x3 - 4x2 + 9x - 6 1 -3 6 1 -3 6 0 Remainder Because the remainder is 0, ƒ112 = 0, and 1 is a zero of the given polynomial function. An x-intercept of the graph of ƒ1x2 = x3 - 4x2 + 9x - 6 is the point 11, 02. The graph in Figure 16 supports this. (b) For ƒ1x2 = x4 + x2 - 3x + 1, remember to use 0 as coefficient for the missing x3-term in the synthetic division. Proposed zero -1)1 0 1 -3 1 -1 1 -2 5 1 -1 2 -5 6 Remainder The remainder is not 0, so -1 is not a zero of ƒ1x2 = x4 + x2 - 3x + 1. In fact, ƒ1-12 = 6, indicating that 1-1, 62 is on the graph of ƒ1x2. The graph in Figure 17 supports this. (c) Use synthetic division and operations with complex numbers to determine whether 1 + 2i is a zero of ƒ1x2 = x4 - 2x3 + 4x2 + 2x - 5. 1 + 2i)1 -2 4 2 -5 1 + 2i -5 -1 - 2i 5 i2 = -1 1 -1 + 2i -1 1 - 2i 0 Remainder The remainder is 0, so 1 + 2i is a zero of the given polynomial function. Notice that 1 + 2i is not a real number zero. Therefore, it is not associated with an x-intercept on the graph of ƒ1x2. S Now Try Exercises 49 and 59. 11 + 2i21-1 + 2i2 = -1 + 4i2 = -5
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