If the leading term of $p$ is $ax^n,a\ne0$, then the leading terms of the equation require $2^nax^{n+1}=8ax^{n+1}$, so $\deg P=3$. Testing, we get that $\boxed{p(x)=c(x-2)(x-4)(x-8)}$, which makes me think that there's a better solution.