Find all polynomials with real coefficients, for which the equality
P(2P(x))=2P(P(x))+2(P(x))2
holds for any real number x.
I started like Steve did, looking at monomials. If p(x)=a*x^n then plug that in, and after simplifying I get a=0 or a=[2^(n-1)-1]^-(n-1).
Most integers n will yield an irrational number, but n=2 simplifies nicely to a=1: p(x)=x^2 is a polynomial that satisfies the given functional equation.
(2x^2)^2 = 2(x^2)^2 + 2(x^2)^2
4x^4 = 2x^4 + 2x^4
This contradicts Steve's assertion that the only possible polynomial for p(x) is the degenerate case when p(x)=0.
Thoughts
