Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.5 - Zeros of Polynomial Functions - Exercise Set - Page 379: 76

Answer

The graph is shown below:
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Work Step by Step

Consider the function: $f\left( x \right)=3{{x}^{4}}+5{{x}^{2}}+2$ The signs of the coefficients of the polynomial $f\left( x \right)$ are: $\begin{matrix} + & + & + \\ \end{matrix}$ Therefore, there is no change in signs of coefficients of $f\left( x \right)$ and this means there are no positive roots. Consider the function: $f\left( -x \right)=3{{x}^{4}}+5{{x}^{2}}+2$ The signs of the coefficients of the polynomial $f\left( -x \right)$ are: $\begin{matrix} + & + & + \\ \end{matrix}$ Therefore, there is no change in signs of $f\left( x \right)$ and this means that there are no negative roots. Therefore, the polynomial does not have any real roots and each of them are complex. Therefore, the function has no positive zeros, no negative zeros, which verify the possible roots obtained using Descartes Rule of Signs.
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