The set P3 is a set of all polynomials of degree 3 or less. In other words, P3 = p(x) = ax3 + bx2 + cx + d, for any a, b, c, d ∈ R}.
(a) What is the zero vector for this vector space? Write it out explicitly.
(b) Show that P3 is a vector space by showing it has the three properties of subspace of all real-valued functions (you’ve already done one property in part (a)).
(c) A basis for P3 is a set of functions that are linearly independent and their span is all of P3. The fundamental idea of a basis means that we can write any polynomial in P3 as a linear combination of the basis vectors. Show that the set B = 1, x, x2, x3 is a basis for P3. (You have two things to show.)