# How do you use the factor theorem to determine whether x-4 is a factor of x^3-21x+20?

The factor theorem states that if $f \left({x}_{0}\right) = 0$, then $\left(x - {x}_{0}\right)$ divides $f \left(x\right)$. So, $\left(x - 4\right)$ divides ${x}^{3} - 21 x + 20$ if and only if the polynomial is zero when $x = 4$.
$f \left(4\right) = {4}^{3} - 21 \cdot 4 + 20 = 64 - 84 + 20 = 0$
So yes, $\left(x - 4\right)$ is a factor of ${x}^{3} - 21 x + 20$. Indeed, the three roots are $- 5$, $1$ and $4$, so we have
${x}^{3} - 21 x + 20 = \left(x - 1\right) \left(x - 4\right) \left(x + 5\right)$