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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 states that the possible number of negative real roots of a polynomial equals the number of sign changes after exchanging the variable \(x\) by \(-x\), or less by an even integer. In the example given, there may be 2 or 0 negative real roots.

Step by step solution

01

Describe Descartes's Rule of Signs

Descartes's Rule of Signs is a method used to determine the possible number of positive and negative real roots of a polynomial. The rule states that the possible number of positive real roots of a polynomial is given by the number of sign changes in the polynomial when written in standard form, or less by an even integer. The possible number of negative real roots is given by the number of sign changes when the signs of the polynomial's coefficients are changed.
02

Application of Descartes's Rule of Signs for Negative Roots

To apply Descartes's Rule of Signs to determine the number of negative real roots, we take the polynomial in standard form and change the signs of the coefficients of all terms. We then count the number of sign changes from term to term. This count gives us the maximum possible number of negative real roots. Furthermore, the actual number of negative real roots will be this count or less by an even integer.
03

Example of Descartes's Rule of Signs

Consider an example polynomial \(P(x) = x^4 - 2x^3 - 2x^2 + 4x + 3\). To find possible negative roots, we change the signs and obtain \(P(-x) = -x^4 + 2x^3 + 2x^2 - 4x -3\). Here, we observe 2 sign changes which implies there could be 2 or (2-2) = 0 negative real roots, according to Descartes's Rule of Signs.

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