# How do you use the factor theorem to determine whether x-4 is a factor of f(x) = x^4 - 12x^2 - 64?

Dec 12, 2015

Use the fact that $x - {x}_{0}$ is a factor of $f \left(x\right)$ if and only if $f \left({x}_{0}\right) = 0$.

#### Explanation:

When you have a polynomial $f \left(x\right)$, if you find a number ${x}_{0}$ such that $f \left({x}_{0}\right) = 0$, then you know that $f \left(x\right)$ can be divided by $x - {x}_{0}$, which means that $f \left(x\right) = \left(x - {x}_{0}\right) g \left(x\right)$, where $g \left(x\right)$ is a polynomial of lower degree.

So, $x - 4$ is a factor of your function if and only if $x = 4$ is a root. Let's check it:

$f \left(4\right) = {4}^{4} - 12 \cdot {4}^{2} - 64 = 256 - 12 \cdot 16 - 64 = 256 - 192 - 64 = 0$