Proving the remainder when a polynomial is divided by an integer. How should I go around proving that $\forall x \in \mathbb{Z}$, the remainder when $x^2+2x$ is divided by $3$ is $0$ or $2$?
Do I use the division algorithm for this one?

Exhaust the possibilities:
When $x\equiv 0$: $x^2+2x\equiv 0+0\equiv 0$.
When $x\equiv 1$: $x^2+2x\equiv 1+2\equiv 0$.
When $x\equiv 2$: $x^2+2x\equiv 4+4\equiv 2$.
