Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial.
What is the value of defining polynomials in this (strange) way? What situations does it make things simpler?
I ask this because it seemed natural to me to define polynomials as a subset of functions, so I was surprised by this.