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Pre-Calc/Trig 3 Descartes Rule of Signs Notes Name __________________________________ A review of finding the zeros of a polynomial using the Rational Root Theorem: Find all the zeros of the polynomial. 1. p( x) x3 2 x 2 4 x 8 2. f ( x) x 4 2 x3 2 x 2 2 x 3 Wouldn’t it be nice to have an idea what the roots will be like before we do the problem? That way we have an idea which ones we should try in the synthetic division. Descartes Rule of Signs p(x) is a polynomial whose terms are arranged in descending powers of a variable 1. The number of positive real zeros of p(x) is the same number as the number of sign changes of the coefficients of terms or is less than this by an even number p( x) x 4 9 x3 24 x 2 6 x 40 2. The number of negative real zeros of p(x) is the same number as the number of sign changes of the coefficients of the terms of p(-x) or is less than this by an even number p( x) ( x)4 9( x)3 24( x)2 6( x) 40 Upper and Lower Bounds Theorem f(x) is a polynomial with real coefficients and a positive leading coefficient. If f(x) is divided by (x-c) using synthetic division, then: 1. If c >0 and each number in the last row is either positive or zero, then c is an upper bound for the real zeros of f(x) 2. If c<0 and the numbers in the last row are alternately positive and negative (zero counts as either pos. or neg.), then c is a lower bound for the real zeros of f(x) Examples State the number of positive real, negative real, and imaginary zeros. 1. f ( x) 5x3 8x 2 4 x 3 2. p( x) 4 x5 3x 4 2 x3 5x 2 6 x 1 3. r( x) x5 6 x 4 3x3 7 x 2 8x 1 Use Descartes Rule of Signs to help you find the roots for the following equations. 1. x3 6 x 2 13x 6 0 2. x 4 7 x3 13x 2 3x 18 0 3. 3x 4 16 x3 7 x 2 64 x 20 0 4. x3 4 x 2 x 26