Question

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

Answer 1

Use the fact that `x-x_0` is a factor of `f(x)` if and only if `f(x_0)=0`.

Explanation:

When you have a polynomial `f(x)`, if you find a number `x_0` such that `f(x_0)=0`, then you know that `f(x)` can be divided by `x-x_0`, which means that `f(x)=(x-x_0)g(x)`, where `g(x)` 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(4)=4^4-12*4^2-64 = 256 - 12*16 - 64 = 256-192-64=0`

So, the answer is yes.