Let V be the subspace of P3 consisting of polynomialspx a

Let V be the subspace of P3, consisting of polynomials,p(x) = a + bx + cx2+dx3 , where p(1) = 0 and p\'(1) = 0. Here p\' represents the derivative of p.

Solution

B.1

p(1)=a+b+c+d=0 , a=-b-c-d

p\'(1)=b+2c+3d=0 , b=-2c-3d

a=2c+3d-c-d=c+2d

p(x)=(c+2d)-(2c+3d)x+cx^2+dx^3 =c(1-2x+x^2)+d(2-3x+x^3)

So basis is

{1-2x+x^2,2-3x+x^3}

B.2

p(1)=1+2-7+4=0

p\'(x)=2-14x+12x^2

p\'(1)=0

So this polynoimal is in B

p(x)=1+2x-7x^2+4x^3

basis is

{1-2x+x^2,2-3x+x^3}

Second vector is only one iwth x^3 so coordinate w.r.t. second vector is 4

So,

1+2x-7x^2+4x^3=a(1-2x+x^2)+4(2-3x+x^3)

1+2x-7x^2=a+8+x(-2a-12)+ax^2

a=-7

So coordinates are: (-7,4)