Question 12
(NET 1) Let $f,g$, and $a$ be polynomials with real coefficients, $f$ and $g$ in one variable and $a$ in two variables. Suppose
$$f(x)-f(y)=a(x,y)(g(x)-g(y))$$
for all $x,y\in\mathbb R$.
Prove that there exists a polynomial $h$ with $f(x)=h(g(x))$ for all $x\in\mathbb R$.
$$f(x)-f(y)=a(x,y)(g(x)-g(y))$$
for all $x,y\in\mathbb R$.
Prove that there exists a polynomial $h$ with $f(x)=h(g(x))$ for all $x\in\mathbb R$.