How can I find all polynomials $f(x)$ such that $f(x)f(x-1)+f(x^2)=0?$ I am self-studying functional equations, but don't know how to start this one. A hint would suffice.
2026-04-03 17:06:49.1775236009
Polynomials $f(x)$ such that $f(x)f(x-1)+f(x^2)=0$
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Hint
Observe that for every $a \in \mathbb{R}$ such that $f(a)=0$, we also have $f(a^2)=0$ and $f((a+1)^2)=0$. This will lead to infinitely many roots unless.....