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: 20


Remainder = $2$ $(x+\frac{1}{3})$ is not a factor of $f(x)$.

Work Step by Step

The Remainder Theorem states that when a function $f(x)$ is divided by $(x-R)$ , then the remainder will be: $f(R)$. We have: $f(x)=3x^4+x^3-3x+1$ $f(\frac{-1}{3})=(3)(\frac{-1}{3})^4+(\frac{-1}{3})^3-(3)(\frac{-1}{3})^3+1=2 \ne 0$ The Factor Theorem states that if $f(a)=0$, then $(x-a)$ is a factor of $f(x)$ and vice versa. Therefore, by the Factor Theorem $(x+\frac{1}{3})$ is not a factor of $f(x)$.
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