5-141 Find all non-zero polynomials $f(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ that satisfy the equation $f\left(x^{2}\right)=(f(x))^{2}$, where $x$ is a real number.
[Solution] Let the $n$-degree polynomial $f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$ satisfy the condition $(f(x))^{2}=f\left(x^{2}\right), x \in R$.
Since the coefficient of $x^{2 n}$ should be equal, we have $a_{n}^{2}=a_{n}$. Because $a_{n} \neq 0$, it follows that $a_{n}=1$.
Similarly, the coefficient of $x^{2 n-1}$ should be equal, so
$$
2 a_{n-1} a_{n}=0 \text{, i.e., } a_{n-1}=0 \text{. }
$$

By the same reasoning, we get $a_{n-2}=\cdots=a_{1}=a_{0}=0$. Therefore, $f(x)=x^{n}$.