22.9. (SRP, 80). Find all non-zero polynomials $P(x)$ that satisfy the identity

$$
P\left(x^{2}\right) \equiv (P(x))^{2}, \quad x \in \mathbf{R}
$$
22.9. Let the desired polynomial have the form

$$
P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \text {, where } a_{n} \neq 0 \text {. }
$$

Assume that at least one of the coefficients $a_{n-1}, \ldots, a_{1}, a_{0}$ is different from zero. Choose the largest value of $k<n$ for which $a_{k} \neq 0$. Then we have

$$
\begin{aligned}
& P\left(x^{2}\right) \equiv a_{n} x^{2 n}+a_{k} x^{2 k}+\ldots+a_{1} x^{2}+a_{0} \equiv \\
&=\left(a_{n} x^{n}+a_{k} x^{k}+\ldots+a_{1} x+a_{0}\right)^{2}=(P(x))^{2} .
\end{aligned}
$$

By comparing the coefficients of $x^{n+h}$, we obtain the equality $0=2 a_{n} a_{k}$, which contradicts the conditions $a_{n} \neq 0, a_{k} \neq 0$. Therefore, $a_{n-1}=\ldots=a_{1}=a_{0}=0$ and $P(x)=a_{n} x^{n}$. Finally, from the conditions $a_{n} x^{2 n}=$ $=P\left(x^{2}\right)=(P(x))^{2} \equiv a_{n}^{2} x^{2 n}$ we get $a_{n}=1$, i.e., $P(x)=x^{n}$, where $n \in \mathrm{Z}^{+}$.