Question

Use the Factor Theorem to determine if $x+3$is a factor of $x^3+8x^2+21x+18$.

Short answer

$x+3$ is a factor of the given polynomial.

Step-by-step solution

Step 1. Given information 

• The given polynomial is $x^3+8x^2+21x+18$.
• The given factor is$x+3$.

Step 2. Applying Factor Theorem

• By factor theorem, if $x+3$is a factor, then the value of the polynomial at $x=-3$will be 0.
• Find the value of the polynomial at $x=-3$.

$\begin{array}{rcl}{\left(-3\right)}^3+8{\left(-3\right)}^2+21\left(-3\right)+18& =& -27+72-63+18\\ & =& 0\end{array}$

• So,$x+3$is a factor of the polynomial$x^3+8x^2+21x+18$.