Proposition. If $f \mid g$, then $g(x)$ splits $\implies$ $f(x)$ splits. [1501]

Let $K / F$ be a field extension and $f(x), g(x) \in F[x]$ such that $f(x) \mid g(x)$. Then $g(x)$ splits completely over $K$ $\implies$ $f(x)$ splits completely over $K$.