The Remainder theorem

The remainder when a polynomial p(x) is divided by x-a is p(a).

The proof is quite simple: if the quotient is q(x) and the remainder is r then by definition of division, p(x)=(x-a)q(x)+r

Now consider this equation at x=a. Substituting gives
p(a)=(a-a)q(a)+r and of course the first term is 0, leaving r=p(a).