34.15. Find all polynomials $P(x)$ for which the identity $x P(x-1)=(x-26) P(x)$ holds.
34.15. For $x=0$ we get $0=-26 P(0)$, i.e. $P(0)=0$. For $x=1$ we get $P(0)=-25 P(1)$, i.e. $P(1)=0$. Next, we set $x=2,3, \ldots, 25$ and sequentially obtain $2 P(1)=-24 P(2), \ldots, 25 P(24)=-P(25)$. Therefore, $P(0)=P(1)=\ldots=P(25)=0$. This means that $P(x)=x(x-1)(x-2) \ldots(x-25) Q(x)$, where $Q(x)$ is some polynomial. From the identity $x P(x-1)=(x-26) P(x)$ it follows that $Q(x-1)=Q(x)$, i.e. $Q(x)=c-$ a constant.