Let $\displaystyle \mathbb{F}$ be a field, let $\displaystyle f \in \mathbb{F}[x]$ be a polynomial, and let $\displaystyle \alpha \in \mathbb{F}$.

Then I understand that if $\displaystyle (x-\alpha) | f$, then $\displaystyle f(\alpha) = 0$, this is obvious.

But it doesn't seem obvious to me that if $\displaystyle f(\alpha)=0$, then $\displaystyle (x-\alpha)|f$. How can I show this? Is there some result about factorisation that implies this?

Any help would be appreciated.