Download Descartes Rule of Signs Notes

Pre-Calc/Trig 3
Descartes Rule of Signs Notes
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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