Precalculus (6th Edition) Blitzer

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

Chapter 2 - Section 2.4 - Dividing Polynomials; Remainder and Factor Theorems - Exercise Set - Page 365: 73


The given statement is false.

Work Step by Step

We know that for a polynomial $f\left( x \right)$ divided by another polynomial $g\left( x \right)$ with quotient obtained $h\left( x \right)$ and remainder $R$, the following relation holds: $f\left( x \right)=g\left( x \right).h\left( x \right)+R$. Here, if $R\ne 0$, $g\left( x \right)$ can never be a factor of $f\left( x \right)$. But when $R=0$: $\begin{align} & f\left( x \right)=g\left( x \right).h\left( x \right)+0 \\ & \frac{f\left( x \right)}{g\left( x \right)}=h\left( x \right) \end{align}$ So, in this case, $g\left( x \right)$ is factor of $f\left( x \right)$. So, the given statement is true for only one whole number $0$ but not any other whole number. Therefore, the given statement is false.
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