# Find all $p\in\mathbb{R}[x]$ such that $p\circ q=q\circ p$ for all $q\in\mathbb{R}[x]$

Find all polynomials $p(x)$ such that $p(q(x)) = q(p(x))$ for every polynomial $q(x)$.

Thanks

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Yes,I have.I have made as far as the answers were given.But wasnt able to proof if more than the solutions i have provided exist! – Sai Krishna Deep Oct 16 '12 at 7:13

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 ... ?

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Yes i got the constant part.But how do we prove if there is a polynomial of some degree greater than 1? How do i see upto what degree it can go, inturn analyze for more solutions. – Sai Krishna Deep Oct 16 '12 at 7:11
@Sai: What I did in the hint tells you exactly what polynomial $p(x)$ has to be, because it tells you the value of $p(x)$ at every real number. – Brian M. Scott Oct 16 '12 at 7:20
Oh! Thanks sir.I missed that part.My bad – Sai Krishna Deep Oct 16 '12 at 7:21
@Sai: You’re welcome. – Brian M. Scott Oct 16 '12 at 7:22