Let q be prime. Show that for any polynomial p(x) with integer coefficients not all divisible by q, p(x) \equiv 0 \bmod q for at \operatorname{most} \operatorname{deg}(p) distinct values of x.
Let q be prime. Show that for any polynomial p(x) with integer coefficients not all divisible by q, p(x) \equiv 0 \bmod q for at \operatorname{most} \operatorname{deg}(p) distinct values of x.