Chapter 3 - Polynomial and Rational Functions - Section 3.2 The Real Zeros of a Polynomial Function - 3.2 Assess Your Understanding - Page 223: 11

Remainder = $8$ $(x-2)$ is not a factor of $f(x)$.

The Remainder Theorem states that when a function $f(x)$ is divided by $(x-R)$ , then the remainder will be: $f(R)$. Now, $f(2)=4(2)^3-3(2)^2-(8)(2)+4=8 \ne0$, 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 $(x-2)$ is not a factor of $f(x)$ .