polynomials and continuity?

Question

Suppose F,G,Q are polynomials and
F(x)/Q(x)=G(x)/Q(x) for all x except when
Q(x)=0. Prove that F(x)=G(x) for ALL x.
OK, I get that this question involves the continuity of polynomials, and I presume that
the limit definition of continuity is to be applied to this problem, but I can't quite get how to apply it. Any pointers would be much appreciated.

Answer 1

Note that when Q(x)≠0, F(x)/Q(x) = G(x)/Q(x) implies that F(x) = G(x). Now, suppose Q(c) = 0. Since F(x)=G(x) on all but a finite set of points (specifically, those points where Q(x) = 0) there is a neighborhood of c in which F(x) = G(x) ∀x≠c. Therefore, [x→c]lim F(x) = [x→c]lim G(x). But since polynomials are continuous, F(c) = [x→c]lim F(x) and G(c) = [x→c]lim G(x), so by transitivity, F(c)=G(c). Repeating this argument for each of the other roots of Q, we find that F(x) = G(x) ∀x. Q.E.D.