Q. 5.91 - Intermediate Algebra

Question

Use the Factor Theorem to determine if $x - 5$ is a factor of $f \left(x\right) = x^3 - 125$.

Step-by-Step Solution

Answer

Yes, $x-5$ is a factor of $f\left(x\right)=x^3-125$

Step 1. Given Information

We want to use the Factor Theorem to determine if $x-5$ is a factor of $f\left(x\right)=x^3-125$

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 − 5 is a factor of $f \left(x\right) = x^3 - 125$ if $f \left(5\right) = 0$

$f \left(x\right) = x^3 - 125$

To evaluate f (5) substitute x = 5.

$f \left(5\right) = 5^3 - 125$

Simplify.

$f \left(5\right) = 125 - 125$

Subtract.

$f \left(5\right) = 0$

Since$f \left(5\right) = 0$, $x - 5$ is a factor of $f \left(x\right) = x^3 - 125$.