How would I go about proving that $\displaystyle \sum_{i=1}^{p-1}a_ix^i \equiv 0 \ (mod \ p) \Rightarrow a_i \equiv 0 \ (mod \ p)$ for all $\displaystyle a_i$.

The way I'm doing it now assumes that there is one coefficient which is not congruent to 0 mod p. This implies that the variable must be congruent to 0 mod p, but since the summation is true for all x, we get our contradiction.

My problem is that I'm assuming that there is only one coefficient which isn't congruent to 0 mod p. In the case of 2 coefficients not congruent to 0, I think I can do it by showing that the two terms have to be additive inverses, and eventually we get back to x being forced to zero (contradiction). What do I do in the case of more than 2 coefficients not being congruent to 0 mod p?