Let $f(x)$ be a polynomial with real coefficients such that $f(0)=1$, $f(2)+f(3)=125$ and $f(x)f(2x^2)=f(2x^3+x)$. Find $f(5)$. (2007 AIME II, Problem 14)
I've tried many ways of solving but I never got to the answer. How do I do it?
Edit: I substituted $x=i$ and found that $f(2)=1$. But I don't know if a polynomial can have a complex domain. Can it?
Also found out that the function is odd (f(-x)=-f(x)) I think.
$f(x)=1+2x^2+x^4$ seems to do the trick.
There is no systematic way to solve an arbitrary functional equation. Given that $f$ is a polynomial, you might want to derive some equations on its coefficients. Or you may just start playing with random polynomials right away. You'll discover pretty soon that $x^2+1$ fits the functional equation, but not the starting conditions. Wait, but then maybe we'll try a square of that?