5. Find all non-zero polynomials $P(x)$ that satisfy $P\left(x^{2}\right) \equiv(P(x))^{2}, x \in \mathbf{R}$.
5. Let the required polynomial be $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$, where $a_{n} \neq 0$. Suppose the coefficients $a_{n-1}, \cdots, a_{1}, a_{0}$ are not all zero, and let $k(<n)$ be the largest integer such that $a_{k} \neq 0$. Then,
$$
P\left(x^{2}\right) \equiv a_{n} x^{2 n}+a_{k} x^{2 k}+\cdots+a_{1} x^{2}+a_{0} \equiv\left(a_{n} x^{n}+a_{k} x^{k}+\cdots+a_{1} x+a_{0}\right)^{2} \equiv(P(x))^{2} \text {, }
$$

By comparing the coefficient of $x^{n+k}$, we get $0=2 a_{n} a_{k}$, which contradicts $a_{n} \neq 0, a_{k} \neq 0$. Therefore, $a_{n-1}=\cdots=a_{1}=a_{0}=0$, and $P(x)=a_{n} x^{n}$. Finally, from the condition $a_{n} x^{2 n} \equiv P\left(x^{2}\right) \equiv(P(x))^{2} \equiv a_{n}^{2} x^{2 n}$, we get $a_{n}=1$, i.e., $P(x)=x^{n}, n \in \mathbf{Z}^{+}$.