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Chapter 2: Problem 34

Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=x^{5}-x^{4}-7 x^{3}+7 x^{2}-12 x-12$$

Short Answer

Expert verified
The polynomial function \(P(x)=x^{5}-x^{4}-7x^{3}+7x^{2}-12x-12\) has either 3 or 1 positive real zero(s) and either 2 or 0 negative real zero(s) according to Descartes' Rule of Signs.

Step by step solution

01

Find the number of positive real zeros

Count the number of sign changes in the polynomial \(P(x)=x^{5}-x^{4}-7x^{3}+7x^{2}-12x-12\). The signs of the terms are positive, negative, negative, positive, negative, and negative. Therefore, there are three sign changes. This means there are either 3 or 3-2=1 positive real zero(s).
02

Find the number of negative real zeros

Substitute \(x\) with \(-x\) to find the sign changes. The polynomial becomes \(P(-x)=(-x)^{5}-(-x)^{4}-7(-x)^{3}+7(-x)^{2}-12(-x)-12=-x^{5}-x^{4}+7x^{3}+7x^{2}+12x-12\). The signs of the terms are negative, negative, positive, positive, positive, and negative. Therefore, there are two sign changes. This means there are either 2 or 2-2=0 negative real zero(s).

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