Let P3= {polynomials in one variable t of degree at most 3} U {0}. (a) Let p(t) = a + bt + ct where a, b, c E R. Compute p'(t) and p'(0). (b) Let H be the set of polynomials p(t) in P3 such that p(0) = 0 and p'(0) = 0. Show that I is a subspace of P3 by verifying the three subspace properties. (c) Let H be the set of polynomials p(t) in P3 such that p(0) = 0 and p'(0) = 0. Show that H is a subspace of P3 by expressing H as a span of finitely many polynomials. (d) Find a basis for H. What is the dimension of H?