Let f() be a polynomial in Zp[z]. If f'(c) = 0, prove that there is some polynomial g(.) in Zplx] such that f(r) = g(r). [Hint: First prove that if f(r) = Ci_0 Cri, then C = 0 if and only if i is a multiple of p. Then use the fact that a^P = a for all a ∈ Zp and the fact that (a + b)^p = a^P + b^P in Zp.] Let F be a field and let a be a root of f(z) ∈ Flx] with multiplicity. Show that a is a root of f'(x) with multiplicity at least C. (We showed in class that the multiplicity was exactly C if F has characteristic 0. The same proof works here with the weaker conclusion.) Prove that if p(z) is an irreducible polynomial in Zplz], then p(r) has no repeated roots. [Hint: If p(z) has a repeated root, use part b) to see that p'(1) and p(z) are not relatively prime. Since p(z) is irreducible, this would force p'(x) = 0. Now apply part 'a) to deduce that p(z) is not irreducible.]