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Chapter 3: Problem 68

Describe how to use Descartes's Rule of Signs to determine the possible number of negative roots of a polynomial equation.

Short Answer

Expert verified
Descartes's Rule of Signs can be used to estimate the number of positive and negative roots in a polynomial equation. Just check the number of sign changes in the standard form of the equation for positive roots. Then replace x with -x in the equation, and again count the sign changes to get the possible number of negative roots.

Step by step solution

01

Descartes鈥檚 Rule of Signs Introduction

Descartes's Rule of Signs is a method used to estimate the number of positive or negative roots of a polynomial. The rule is based on the number of times the sign (+ or -) changes in the polynomial equation when expressed in standard form from highest degree to lowest.
02

Write down the polynomial in standard form

The first step is to organize the polynomial in standard form, which means to write it in descending order, starting from the term with the highest degree to the term with the lowest degree.
03

Count the sign changes for positive roots

Next, count the number of times the sign changes between terms when written in standard form. Each sign change indicates a possible positive root.
04

Substitute x = -x for the variable in the equation

To find the possible number of negative roots, the value of x in the polynomial equation is replaced by -x. The equation then goes through a transformation each term with an odd degree changes sign, and each term with an even degree maintains the sign.
05

Count the sign changes for negative roots

In this transformed equation, again count the number of times the sign changes between the terms. This gives the possible number of negative roots for the polynomial equation.

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