Find all polynomials such that

$$
(X-16) P(2 X)=16(X-1) P(X)
$$
Let's first find the degree. We assume that $P$ is non-zero.

Direct calculation of the degree does not help us. We look at the leading coefficient, with $n$ being the degree and $\lambda$ the leading coefficient:

$$
1 \cdot 2^{n} \lambda=16 \lambda
$$

Thus, $n=4$.

By evaluating at 1, we have $P(2)=0$, then we re-evaluate at 2, 4, and 8, to get $P(4)=P(8)=$ $P(16)=0$. By the degree, $P(X)=\lambda(X-2)(X-4)(X-8)(X-16)$.

These polynomials work (for $\lambda \in \mathbb{R}$).