Negative integers and polynomial congruence classes

Question

Let's take a polynomial $m(x)$ from $\mathbb{Z}_{3}[x]$. Now, $\mathbb{Z}_{3}$ should contains the integers $-1,-2,-3$. However after reading few exercises about this argument i suspect that we can ignore negative values when we work modulo $m(x)$. But, why ? (Intuitively)

Answer 1

$\mathbb Z_3=\{[0],[1],[2]\}$ where $$[n]=\{n+3k|k\in\mathbb Z\}$$ which means that $[-3]=[0], [-2]=[1]$ and $[-1]=[2]$.