224 CHAPTER 4 Polynomial and Rational Functions Obtaining Graphs Using Bounds on Zeros Obtain a graph for the polynomial function. f x x x x 2 11 7 6 3 2 ( ) = + − − EXAMPLE 9 Solution Based on Example 8, every zero lies between 7− and 1. Using Xmin 7 = − and Xmax 1, = we graph Y f x x x x 2 11 7 6 1 3 2 ( ) = = + − − . Figure 30 shows the graph on a TI-84 Plus CE obtained using ZOOM-FIT. Now Work PROBLEM 81 Figure 30 70 2104 27 1 The next example shows how to proceed when some of the coefficients of the polynomial are not integers. (c) 5 25 3 4 (a) Figure 31 3000 21500 25 6 (b) 260 220 25 6 Finding the Real Zeros of a Polynomial Function Find all the real zeros of the polynomial function f x x x x x x 1.8 17.78 31.61 37.9 8.7 5 4 3 2 ( ) = − − + + − Round answers to two decimal places. Solution EXAMPLE 10 Step 1 There are at most five real zeros. There are three positive zeros or one positive zero, and there are two negative zeros or no negative zeros. Step 2 Since there are noninteger coefficients, the Rational Zeros Theorem does not apply. Step 3 Determine the bounds on the zeros of f. Using synthetic division with successive integers, beginning with 1, ± a lower bound is 5− and an upper bound is 6. Every real zero of f lies between 5− and 6. Figure 31(a) shows the graph of f using a TI-84 Plus CE with Xmin 5 = − and Xmax 6 = . Figure 31(b) shows a graph of f after adjusting the viewing window to improve the graph. Step 4 From Figure 31(b), we see that f appears to have four x-intercepts: one near 4, − one near 1, − one between 0 and 1, and one near 3.The x-intercept near 3 might be a zero of even multiplicity since the graph seems to touch the x-axis at that point.
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