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Use Descartes' rule of signs to describe the roots for each polynomial function 1....

Question:

Use Descartes' rule of signs to describe the roots for each polynomial function

{eq}1. m(x)=x^3+3x^2-18x-40 \\2. w(x) = x^4+4x^3-7x^2+20x-21 {/eq}

Descartes' Rule of Signs

Descartes' Rule of Signs is a quick trick to giving an idea of how many positive real roots a polynomial has and how many negative real roots a polynomial has. Descartes' Rule of Signs involves counting the number of sign changes in a polynomial function - where, by sign change we mean changing from "+" to "-" or vice versa in consecutive terms. The Rule of Signs states that :

  • The number of positive real roots of a polynomial {eq}f(x) {/eq} is equal to the number of sign changes of {eq}f(x) {/eq}, or less than that number by an even integer.
  • The number of negative real roots of a polynomial {eq}f(x) {/eq} is equal to the number of sign changes of {eq}f(-x) {/eq}, or less than that number by an even integer.

Answer and Explanation:

1. Using Descartes' Rule of Signs for {eq}m(x)=x^3+3x^2-18x-40 {/eq} we must count the number of sign changes.

There is only one sign change in {eq}m(x) {/eq} and so there must be 1 positive real root of the polynomial.

For the negative real roots, we must count the sign changes for {eq}m(-x) = (-x)^3 + 3(-x)^2 - 18(-x) - 40\\ m(-x) = -x^3 + 3x^2 + 18x - 40 {/eq}

There are two sign changes in {eq}m(-x) {/eq} and so there is either 2 negative real roots or 0 negative real roots of the polynomial.


2. Using Descartes' Rule of Signs for {eq}w(x) = x^4 + 4x^3 - 7x^2 + 20x -21 {/eq} we must count the number of sign changes.

There are three sign changes in {eq}w(x) {/eq} and so there must be either 3 or 1 positive real root of the polynomial.

For the negative real roots, we must count the sign changes for {eq}w(-x) =(-x)^4 + 4(-x)^3 - 7(-x)^2 + 20(-x) - 21\\ w(-x) = x^4 -4x^3 - 7x^2 -20x - 21 {/eq}

There is only one sign change in {eq}w(-x) {/eq} and so we must have one negative real root of the polynomial.


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