SECTION 4.3 The Real Zeros of a Polynomial Function 217 (b) Figure 27 CAUTION Remember that in order for synthetic division to be used, the divisor must be of the form −x c . j THEOREM Number of Real Zeros A polynomial function cannot have more real zeros than its degree. Proof The proof is based on the Factor Theorem. If r is a real zero of a polynomial function f , then f r 0 ( ) = , and x r − is a factor of f x( ) . Each real zero corresponds to a factor of degree 1. Because f cannot have more first-degree factors than its degree, the result follows. ■ THEOREM Descartes’ Rule of Signs Suppose f is a polynomial function written in standard form. • The number of positive real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f x( ) or else equals that number less an even integer. • The number of negative real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f x ( ) − or else equals that number less an even integer. Figure 27(b) shows that ( ) − =− f 2 27 using Desmos. Because ( ) − =− ≠ f 2 27 0, conclude from the Factor Theorem that x x 2 2 ( ) − − = + is not a factor of f x( ) . From Example 2(a), x 1 − is a factor of f . To write f in factored form, divide f by x 1 ( ) − using synthetic division. ) − − 1 2 1 2 3 21 3 2 13 0 The quotient is q x x x 2 3 2 ( ) = + + with a remainder of 0, as expected. Write f in factored form as f x x x x x x x 2 2 3 1 2 3 3 2 2 ( ) ( ) ( ) = − + − = − + + But how many real zeros can a polynomial function have? In counting the zeros of a polynomial, count each zero as many times as its multiplicity. Now Work PROBLEM 11 2 Use Descartes’ Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function Descartes’ Rule of Signs provides information about the number and location of the real zeros of a polynomial function written in standard form (omitting terms with a 0 coefficient). It uses the number of variations in the sign of the coefficients of f x( ) and f x ( ) − . For example, the following polynomial function has two variations in the signs of the coefficients. f x x x x x 3 4 3 2 1 7 4 2 ( ) = − + + − − to − + to + − Replacing x by x− gives ( ) ( ) ( ) ( ) ( ) −=−− +− +− −−− = + + + − f x x x x x x x x x 3 4 3 2 1 3 4 3 2 1 7 4 2 7 4 2 to + − which has one variation in sign. We do not prove Descartes’ Rule of Signs. Let’s see how it is used.
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