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Chapter 2: Problem 67

Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

Short Answer

Expert verified
To determine the number of positive real roots of a polynomial, we use Descartes's Rule of Signs. First, write the polynomial in standard form. Then, count the number of sign changes. The number of positive real roots of the polynomial is either equal to this count, or less by a multiple of 2.

Step by step solution

01

Write the Polynomial in Standard Form

If the polynomial is not already in standard form it must be rearranged so it is. The standard form of a polynomial lists terms by degree from highest to lowest, for example \(f(x) = x^4 - 2x^3 + 3x^2 - x - 3\).
02

Count the number of Sign Changes

A sign change occurs every time a term's coefficient switches from positive to negative or negative to positive as you read from left to right. Each switch marks a possible positive real root, for example with the polynomial from Step 1 there are 3 sign changes: \(x^4\) (positive) to \(-2x^3\) (negative), \(3x^2\) (positive) to \(-x\) (negative), and \(-x\) (negative) to \(-3\) (positive).
03

Apply Descartes Rule

According to Descartes鈥檚 Rule of Signs, the number of positive real roots is either equal to the number of sign changes, or less than this number by a multiple of 2. Thus if there are 3 sign changes, there may be 3, 1 or no positive real root.

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