I have already shown that any polynomial $P\in\mathbb{R}[x]$ satisfies $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$
My question is, given a polynnomial $P\in\mathbb{C}[x]$, how I can verify whether $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}\Rightarrow P\in\mathbb{R}[x]$ is always true?
So, does there exist a polynomial $P\in\mathbb{C}[x]\setminus\mathbb{R}[x]$ such that $\forall x\in\mathbb{C}:\overline{P(\overline{x})}=P(x)$?