Just by observation,I found that p(x)=x satisfies the given eqn. and also that p(0)=0.

So,we have,p'(x)=1 for all x.Hence,p'(0)=1.

But I don't know whether p(x)=x is the only polynomial which satisfies the given eqn.and if it is true,we have to prove it.Let us wait for someone to post the solution for this in a more convincing way.Anyway,Is my answer correct?