218 CHAPTER 4 Polynomial and Rational Functions 3 Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function The next result, called the Rational Zeros Theorem , provides information about the rational zeros of a polynomial with integer coefficients . Solution Because the polynomial is of degree 7, by the Number of Real Zeros Theorem there are at most seven real zeros. Since there are three variations in the sign of the nonzero coefficients of f x( ) , by Descartes’ Rule of Signs we expect either three positive real zeros or one positive real zero. To continue, look at f x ( ) − . f x x x x x x 3 4 3 2 3 7 4 3 2 ( ) − = − − − + + − There are two variations in sign, so we expect either two negative real zeros or no negative real zeros. Equivalently, we now know that the graph of f has either three positive x -intercepts or one positive x -intercept and two negative x -intercepts or no negative x -intercepts. Using the Number of Real Zeros Theorem and Descartes’ Rule of Signs Discuss the real zeros of f x x x x x x 3 4 3 2 3 7 4 3 2 ( ) = − + + − − . EXAMPLE 3 Now Work PROBLEM 21 Listing Potential Rational Zeros List the potential rational zeros of f x x x x 2 11 7 6 3 2 ( ) = + − − Solution EXAMPLE 4 Because f has integer coefficients, the Rational Zeros Theorem may be used. First, list all the integers p that are factors of the constant term a 6 0 = − and all the integers q that are factors of the leading coefficient a 2 3 = . p: 1, 2, 3, 6 ± ± ± ± Factors of 6− q: 1, 2 ± ± Factors of 2 Now form all possible ratios p q . p q : 1 1 , 2 1 , 3 1 , 6 1 , 1 2 , 2 2 , 3 2 , 6 2 ± ± ± ± ± ± ± ± which simplify to ± ± ± ± ± ± p q : 1, 2, 3, 6, 1 2 , 3 2 If f has a rational zero, it will be found in this list of 12 possibilities. THEOREM Rational Zeros Theorem Let f be a polynomial function of degree 1 or higher of the form ( ) = + + + + ≠ ≠ − − f x a x a x a x a a a 0 0 n n n n n 1 1 1 0 0 where each coefficient is an integer. If p q , in lowest terms, is a rational zero of f , then p must be a factor of a ,0 and q must be a factor of an . Now Work PROBLEM 33
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