Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 3 - Polynomial and Rational Functions - Section 3.2 The Real Zeros of a Polynomial Function - 3.2 Assess Your Understanding - Page 224: 24

Answer

The maximum number of real zeros is $5$. The number of positive real zeros is $1$. The number of negative real zeros is $0$.

Work Step by Step

We should remember that the maximum number of zeros of a polynomial function $f(x)$ cannot be greater than its degree. We will consider Descartes' Rule of Signs for explaining this solution: (1) The number of negative real zeros of a polynomial function $f(x)$ is either equal to the number of variations in the sign of the non-zero coefficients of $f(−x)$ or that number minus an even integer. (2) The number of positive real zeros of a polynomial function $f(x)$ is either equal to the number of variations in the sign of the non-zero coefficients of $f(x)$ or that number minus an even integer. We can notice from the given polynomial function that the highest degree is $5$, so the maximum number of real zeros is $5$. $f(x)=-3x^5+4x^4+2$ has 1 variation in the sign. So, the number of positive real zeros is $1$. $f(−x)=3x^5+4x^4+2$ has 0 variations in the sign. So, the number of negative real zeros is $0$.
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