Question

Let P3 be the vector space of all polynomials with real coefficients of degree at most 3.

a) Show that the set E={p(x) ∈ P3 : p(0)=p(1)=0} is a subspace of P3.

b) Show that if p ∈ E, then p can be expressed in the following form:

p(x)=x(x-1)(ax+b) : a,b ∈ R

Hence, find a basis for E and dim(E)