Chapter 3 - Polynomial and Rational Functions - Section 3.2 The Real Zeros of a Polynomial Function - 3.2 Assess Your Understanding - Page 224: 15

Remainder = $0$ $(x+3)$ is 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, $f(-3)=(3)(-3)^6+82(-3)^3+27= (3)(729)+(82)(-27)+27=0$ 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 $(x+3)$ is a factor of $f(x)$ .