a) Find all polynomials $p(x)$ such that $p(q(x)) = q(p(x))$ for every polynomial $q(x)$.
b) Find all polynomials $p(x), q(x)$ such that $p(q(x)) = q(p(x))$.
Thanks
a) Find all polynomials $p(x)$ such that $p(q(x)) = q(p(x))$ for every polynomial $q(x)$.
b) Find all polynomials $p(x), q(x)$ such that $p(q(x)) = q(p(x))$.
Thanks
HINT: Consider what happens if $q(x)$ is a constant polynomial, say $q(x)=a$: $p(a)=p\big(q(x)\big)=q\big(p(x)\big)=a\;.$
But there’s a constant polynomial for every $a\in\Bbb R$, so ... ?