 Solution

Divide $f$ by $m$ in the polynomial ring over the field: $f=qm+r$, where $r=0$ or $\mathrm{deg}r<\mathrm{deg}m$. If $f(a)=0$, then $r(a)=0$, and minimality forces $r=0$. Conversely a multiple of $m$ vanishes at $a$. ThereforeThis includes the zero polynomial.