Let $f \in \mathbb{Z}_p [x]$, where $p$ is prime, be defined by $f(x) = x^p - x$. Show that its Polynomial evaluation is identically zero.

Question

I understand that Fermat's Little Theorem is crucial here, but I am not sure how to tie the whole thing together...

Observe: If $f(a) \equiv 0$ then $a^p - a \equiv 0$ for $a \in \mathbb{Z}_p$ $\implies a^p \equiv a\mod p$

So, proof: Since $p$ is prime, then $a^p \equiv a\mod p$ $\implies a^p - a \equiv 0 \mod p$ $\implies f(a) \equiv 0$ for $a \in \mathbb{Z}_p$.

Is this enough or am I incorrectly assuming something?