Q. 5.92 - Intermediate Algebra

Question

Use the Factor Theorem to determine if $x - 6$ is a factor of $f \left(x\right) = x^3 - 216$.

Step-by-Step Solution

Answer

Yes, $x-6$ is a factor of $f\left(x\right)=x^3-216$

Step 1. Given Information

We want to use the Factor Theorem to determine if $x-6$ is factor of $f\left(x\right)=x^3-216$

We know that Factor Theorem:

For any polynomial function f (x),

• if x − c is a factor of f (x), then f (c) = 0

• if f (c) = 0, then x − c is a factor of f (x)

Step 2. Solve

The Factor Theorem tells us that x − 6 is a factor of $f\left(x\right)=x^3-216 if f\left(6\right)=0$

$f\left(x\right)=x^3-216$

To evaluate f (6) substitute x = 6. 

$f\left(6\right)=6^3-216$

Simplify.

$f\left(6\right)=216-216$

Subtract.

$f\left(6\right)=0$

Since $f\left(6\right)=0, x-6$ is a factor of $f\left(x\right)=x^3-216$