Hello. Have a few questions about polynomials that were never really cleared up in my algebra courses.

1. Are a polynomial's derivatives and antiderivatives (of all orders) also polynomials?

2. If a function is equal to a polynomial, but isn't in the ordinary form of one (as in, it looks nothing like the usual sum of powers with coefficients), is it still considered a polynomial function?

3. The proof of the rational root theorem assumes that and are coprime. Does this need to be taken into account when evaluating the roots this way? It never seems I've had to worry about it when simply using it, so why is it necessary in the elementary proof?