TITLE: prove that polynomials can only take $0$ value
QUESTION [2 upvotes]: Let $f, g, h$ be polynomials of real coefficients such that
$$f^2(x) = x g^2(x) + x h^2(x).$$
Prove that $f=g=h=0$.
Tried so far:
Set $x$ to be any negative number then $f^2=-(g^2+h^2)$ times a positive number, this can only hold if $f=g=h=0$.
Or $x=0$, $f=g=h=0$.
But what about positive $x$? There should be some non-zero real $f,g$ or $h$ right? How to show only $0$ is possible?
Thanks!

REPLY [3 votes]: $f,g$ and $h$ have infinitely many zeros. Since they are polynomials thyy have to be identically $0$.
A non-zero polynomial can have only finite number of zeros.
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