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Chapter 2: Problem 31

Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4$$

Short Answer

Expert verified
The function \(P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4\) may have 1 positive real zero and either 3 or 1 negative real zeros.

Step by step solution

01

Find the Possible Number of Positive Zeros

First, count the number of times the sign changes in the polynomial \(P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4\). The sign changes one time. Therefore, the possible number of positive real zeros is one.
02

Find the Possible Number of Negative Zeros

To find the number of negative real zeros, replace each \(x\) with \(-x\) in the polynomial to get \(P(-x) = 6(-x)^{4}+23(-x)^{3}+19(-x)^{2}-8(-x)-4\), which simplifies to \(P(-x)=6x^{4}-23x^{3}+19x^{2}+8x-4\). Count the number of sign changes, there are three sign changes. Thus, the possible number of negative real zeros is 3 or 1.

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