I can't imagine it being too long if you can see what to do...How long do you reckon it would take to multiply out the whole of the RHS and simplify?
Nice way of looking at it! Of course the direct proof of the recursive relation is just summing the equation (x^k)P(x)=0 over the roots of P, where k is a natural number. Would rep but need to share it apparently...I just realised, we don't really need to prove the recursive relationship (do we?). We can see by inspection that its Characteristic Polynomial is just P(x) !