Let a(x, y) be the polynomial x^{2}y + xy^{2}, and b(x, y) the polynomial x^{2} + xy + y^{2}. Prove that we can find a polynomial p_{n}(a, b) which is identically equal to (x + y)^{n} + (-1)^{n} (x^{n} + y^{n}). For example, p_{4}(a, b) = 2b^{2}.

**Solution**

Let us write E(n) = (x + y)^{n} + (-1)^{n} (x^{n} + y^{n}). We use induction on n. We have E(1) = 0, E(2) = 2b, E(3) = 3a.

We find that (x + y)^{n+3} = b (x + y)^{n+1} + a (x + y)^{n} and x^{n+3} + y^{n+3} = b (x^{n+1} + y^{n+1}) - a (x^{n} + y^{n}). So it follows that E(n+3) = b E(n+1) + a E(n). That completes the induction.

© John Scholes

jscholes@kalva.demon.co.uk

23 Jan 2001