Prove the following polynomial equation has no answers

Question

Assume there two polynomials for which the following equation$$P(Q(x))=Q(P(x))$$holds for $x\in\Bbb R$. Also the equation $P(x)=Q(x)$ has no real root. Prove that the following equation$$P(P(x))=Q(Q(x))$$has no real roots either.

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Answer 1

Since $P(x) = Q(x)$ has no real root, one of $P,Q$ must be strictly greater than the other, for all $x \in \mathbb{R}$.

Without loss of generality, assume $P$ is the greater one.

Then for all $x \in \mathbb{R}$, $$P(P(x)) > Q(P(x)) = P(Q(x)) > Q(Q(x))$$