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Chapter 2: 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
The Descartes's Rule of Signs is applied on a polynomial in standard form by first replacing \(x\) with \(-x\) for determining possible negative roots. Then total number of times the signs change is counted. For example, for a polynomial \(2x^4 - 3x^3 + 2x^2 - x - 1\), the possible number of negative roots are 2 or 0. The count can decrease by multiples of 2 due to presence of pairs of complex roots.

Step by step solution

01

Introduction to the Rule

Descartes's Rule of Signs is a technique used to predict the number of positive and negative real roots of a polynomial. It's not about finding the actual roots, but their possible quantity and sign. The rule focuses only on the variations in the sign of the coefficients.
02

Write Polynomial in Standard Form

First thing is to ensure the polynomial equation is written in the standard form where powers are in descending order. Assume for example a polynomial equation \( P(x) = 2x^4 - 3x^3 + 2x^2 - x - 1 \).
03

Apply Rule for Negative Roots

For negative roots, replace \(x\) in the polynomial with \(-x\) and then write out the polynomial. So, \( P(-x) = 2(-x)^4 - 3(-x)^3 + 2(-x)^2 - (-x) - 1 = 2x^4 + 3x^3 + 2x^2 + x - 1 \).
04

Count Number of Sign Changes

Count the number of times the coefficients change sign. From the example in earlier step, the signs changes 2 times: from positive to negative to positive, thus we have 2 or 0 (subtracted by multiples of 2 until 0 or a negative number is achieved) possible negative roots.
05

Explanation for Zeroing Out

The reason the count can decrease by multiples of 2 is due to complex roots. Complex roots come in pairs, so if there's one pair (two complex roots), it will reduce the count of real roots by 2 (because they were counted as sign changes, but aren't actually real roots).

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