Q.334. - Intermediate Algebra

Question

Use the factor theorem to determine whether $x+3$ a factor of $x^3+8x^2+21x+18$.

Step-by-Step Solution

Answer

Yes, $x+3$ is a factor of $x^3+8x^2+21x+18$.

Step 1. Given information.

$f\left(x\right)=x^3+8x^2+21x+18$ and $x+3$ is the factor of polynomial function.

Step 2. To simplify the polynomial function, by using the remainder theorem.

According to the remainder theorem, $x+3$ is a factor of $f\left(x\right)=x^3+8x^2+21x+18$ if $f\left(-3\right)=0$.

Substitute $x=-3$ in $f\left(x\right)=x^3+8x^2+21x+18$.

$$\begin{gathered} f\left(-3\right)={\left(-3\right)}^3+8{\left(-3\right)}^2+21\left(-3\right)+18 \\ \Rightarrow -27+8·9-63+18 \\ \Rightarrow -27+72-63+18 \\ \Rightarrow 90-90 \\ \Rightarrow 0 \end{gathered}$$

Step 3. Simplified answer.

Since $f\left(-3\right)=0$.

Hence, $x+3$ is a factor of $f\left(x\right)=x^3+8x^2+21x+18$.