SECTION 4.3 The Real Zeros of a Polynomial Function 219 In Words For the polynomial function ( ) = + − − f x x x x 2 11 7 6 3 2 , we know 5 is not a zero, because 5 is not in the list of potential rational zeros. However, −1 may or may not be a zero. Be sure that you understand what the Rational Zeros Theorem says: For a polynomial function with integer coefficients, if there is a rational zero, it is one of those listed. It may be the case that the function does not have any rational zeros. The Rational Zeros Theorem provides a list of potential rational zeros of a function f . If we graph f , we can get a better sense of the location of the x -intercepts and test to see if they are rational. We can also use the potential rational zeros to select our initial viewing window to graph f and then adjust the window based on the results.The graphs shown throughout the text will be those obtained after setting the final viewing window. How to Find the Real Zeros of a Polynomial Function Find the real zeros of the polynomial function f x x x x 2 11 7 6 3 2 ( ) = + − − . Write f in factored form. EXAMPLE 5 Figure 28 Y x x x 2 11 7 6 1 3 2 = + − − 70 210 27 3 4 Find the Real Zeros of a Polynomial Function Step-by-Step Solution Step 1 Determine the maximum number of zeros. Also determine the number of positive and negative real zeros. Since f is a polynomial function of degree 3, there are at most three real zeros. From Descartes’ Rule of Signs, there is one positive real zero. Also, since f x x x x 2 11 7 6, 3 2 ( ) − = − + + − there are two negative real zeros or no negative real zeros. From our list of potential rational zeros, we test 6− to determine if it is a zero of f . Because f 6 2 6 11 6 7 6 6 2 216 1136 42 6 432 396 36 0 3 2 ( ) ( ) ( ) ( ) ( ) ( ) − = − + − − − − = − + + − =− + + = we know that 6− is a zero and x x 6 6 ( ) − − = + is a factor of f . Use long division or synthetic division to factor f . (We will not show the division here, but you are encouraged to verify the results shown.) After dividing f by x 6 + , the quotient is x x 2 1 2 − − , so f x x x x x x x 2 11 7 6 6 2 1 3 2 2 ( ) ( ) ( ) = + − − = + − − Step 2 If the polynomial function has integer coefficients, use the Rational Zeros Theorem to identify those rational numbers that potentially can be zeros. Step 3 Using a graphing utility, graph the polynomial function. Step 4 Use the Factor Theorem to determine if the potential rational zero is a zero. If it is, use synthetic division or long division to factor the polynomial function. Repeat Step 4 until all the zeros of the polynomial function have been identified and the polynomial function is completely factored. List the potential rational zeros obtained in Example 4: 1, 2, 3, 6, 1 2 , 3 2 ± ± ± ± ± ± Figure 28 shows the graph of f using a TI-84 Plus CE graphing calculator. We see that f has three zeros: one near 6− , one between 1− and 0, and one near 1. (continued)
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