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.

Sorry for having no idea with this! Do you have any? Please subscribe...

Thanks in advance!

quasi · Accepted Answer · 2018-01-29T02:57:01.763

6

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

edited Jan 29 '18 at 02:57

answered Jan 28 '18 at 22:30

quasi

58,772

Very cute!..... – Mostafa Ayaz Jan 28 '18 at 22:31
NIce. Final paren missing. – marty cohen Jan 29 '18 at 02:53
@marty cohen: Thanks, fixed. – quasi Jan 29 '18 at 02:57

Prove the following polynomial equation has no answers

1 Answers1

Linked

Related