Answer to Question #44929 in Linear Algebra for Anurodh

Question

Let P^3 ={ax^3+bx^2+cx+d ! a,b,c,d ϵ R}. Check whether f (x) = x^2+2x+1 is in[S],
the subspace of P^3 generated by S ={3x^2+1, 2x^2+x+1}.

If f (x) is in [S], write f as a linear combination of elements in S.

If f (x) is not in [S], find anotherpolynomial g(x) of degree at most two such that f (x)
is in the span of S U {g(x)}.

Alsowrite f as a linear combination of elements in S U {g(x)}.