5. 137 Find all polynomials $P(x)$ satisfying $P(0)=0$ and
$$P(x) \equiv \frac{1}{2}(P(x+1)+P(x-1)), x \in R$$
[Solution] The polynomial $P(x)=a x$ satisfies the conditions in the problem, where $a$ is a constant. We will prove this by induction for $n$ $\in Z^{+}$.

The polynomial $P(x)$ satisfies $P(n)=n P(1)$. When $n=0$ and $n=1$, the equation is obviously true. Suppose the equation holds for $n-1$ and $n$, where $n \in N$, then
$$\begin{aligned}
P(n+1) & =2 P(n)-P(n-1) \\
& =2 n P(1)-(n-1) P(1) \\
& =(n+1) P(1)
\end{aligned}$$

Thus, the equation also holds for $n+1$. Therefore, the polynomial $P(x)-P(1) x$ has infinitely many roots $x=0,1,2, \cdots$. Hence, it must be identically zero. So the polynomial we seek has the form $P(x)=$ $a x$.