Let F be a field. In the polynomial ring F[x], let I be all the f(x) for which f(0) = f(1). Show that this I is an ideal in F[x], and find a generator for it -- that is, a polynomial h(x) such that our I consists of all multiple of h(x).

I have absolutely no idea what to do here. How do you show that I is an ideal? What does that even mean? The things we have done in class for it haven't made any sense.