Let W p x elementof P3 p 1 0 Show that W is a subspace of

Let W = {p (x) elementof P_3: p\' (1) = 0}. Show that W is a subspace of P_3. Find a basis for W.

Solution

P(x) belongs to P3 and p\'(1) = 0

So P is of the form W = k* (x-1)^2 * (ax+b)

P\'(x) = 2* (x-1) * k* (ax+b) + k*(x-2)^2 * a

So p\'(1) = 0

So W = k*(x-1)^2 *(ax+b) is a 3 rd degree polynomial satisfying all the vector space properties.

I.e addition and scalar multiplication. SO it is a subspace in P3

(b) Basis of W = {x^3 , x^2 ,x,1 }