220 CHAPTER 4 Polynomial and Rational Functions Factor f completely as follows: ( ) ( ) ( ) ( )( )( ) = + −−=+ −−=+ + − f x x x x x x x x x x 2 11 7 6 6 2 1 6 2 1 1 3 2 2 Notice that the three zeros of f are in the list of potential rational zeros in Step 2, and confirm what was expected from Descartes’ Rule of Signs. Now any solution of the equation x x 2 1 0 2 − − = will be a zero of f . Because of this, the equation x x 2 1 0 2 − − = is called a depressed equation of f . Because any solution to the equation x x 2 1 0 2 − − = is a zero of f , work with the depressed equation to find the remaining zeros of f . The depressed equation x x 2 1 0 2 − − = is a quadratic equation with discriminant b ac 4 1 4 2 1 9 0 2 2 ( ) ( )( ) − = − − − = > . The equation has two real solutions, which can be found by factoring. ( )( ) − − = + − = + = − = = − = x x x x x x x x 2 1 2 1 1 0 2 1 0 or 1 0 1 2 or 1 2 The zeros of f are 6, 1 2 − − , and 1. SUMMARY Steps for Finding the Real Zeros of a Polynomial Function Step 1 Use the degree of the polynomial function to determine the maximum number of zeros. Use Descartes’ Rule of Signs to determine the number of positive and negative real zeros. 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 Graph the polynomial function using a graphing utility to find the best choice of potential rational zeros to test. 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. Each time that a zero (and thus a factor) is found, repeat Step 4 on the depressed equation. In attempting to find the zeros, remember to use (if possible) the factoring techniques that you already know (special products, factoring by grouping, and so on). Finding the Real Zeros of a Polynomial Function Find the real zeros of f x x x x x x 4 16 37 84 84 6 5 3 2 ( ) = + − − − − . Write f in factored form. Solution EXAMPLE 6 Step 1 There are at most six real zeros. There is one positive real zero and there are five, three, or one negative real zeros. Step 2 To obtain the list of potential rational zeros, write the factors p of a 84 0 = − and the factors q of the leading coefficient a 1 6 = . ±±±±±±± ± ± ± ± ± ± p q : 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 : 1 The potential rational zeros consist of all possible quotients p q : ±±±±±±± ± ± ± ± ± p q : 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
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