Solution
= Solution
Because $f$ is <irreducible polynomial>[irreducible], the ideal $(f)$ is maximal in the <polynomial ring> $k[X]$. Hence
$$
F=k[X]/(f)
$$
is a <field extension> of $k$. Let $\alpha=X+(f)\in F$. Evaluation in the quotient gives $f(\alpha)=0$, and the <factor theorem> in $F[X]$ gives
$$
\boxed{f(X)=(X-\alpha)g(X)}
$$
for some $g\in F[X]$. This is the standard <adjoining a root of an irreducible polynomial> construction.