Let Px stand for the real vector space of all polynomials in

Let P[x] stand for the real vector space of all polynomials in x with real coefficients. Show that the set E = {p(x) epsilon R|x] : p(-1) = 0 and deg(p(x)) less than or equal to 3} is a subspace of P|x]. Show that the set {(x + 1),(x +)^2,(x +1)^3} is a basis of E.

Solution

A generalised polynomial of degree 3 is p (x) = r(x + a)³ + s ( x + b)² + t (x + c) + d where a, b, c, d r,s,t are arbitrary constants. If p (-1) = 0, then a = b = c = 1 and d = 0, so that the polynomial changes to p(x) = r (x + 1)³ + r (x +1)² + t ( x + 1). We will first show that E = { p(x) R(x) : p ( -1) = 0 and deg( p(x)) 3 } is a vector space.

Therefore E is a vector space. More specifically, E is a subspace of P [x].

We have seen that any arbitrary element of E is of the form p(x) = r (x + 1)³ + r (x +1)² + t ( x + 1). This means that the elements of E are linear combinations of (x + 1), (x + 1)² and ( x + 1)³ . Therefore, the set { (x + 1), (x + 1)² , ( x + 1)³ } forms a basis of E.