Determining the subspace of a given vectorspace

Question

Given a vectorspace $\mathbb{R}[X]_{<4}$ of the polynomials of 3rd degree with real coefficients. Suppose W = { P(X) $\in$ $\mathbb{R}[X]_{<4}$ | P'(0) = 0}:

1. Prove that W is a subspace of $\mathbb{R}[X]_{<4}$.
2. Determine the dimension of W.
3. Find a subspace W' of $\mathbb{R}[X]_{<4}$ such that $\mathbb{R}[X]_{<4}$ = W $\bigoplus$ W'.

I know how to prove that W is a subspace of $\mathbb{R}[X]_{<4}$, but I don't know to solve the 2nd and the 3rd questions. Thanks for your help.

Answer 1

For point 2

• $p(x)=ax^3+bx^2+cx+d\implies p'(x)=3ax^2+2bx+c \quad p'(x)=0 \implies c=0$

thus

• $p(x)=ax^3+bx^2+d$

with basis $1,x^2,x^3$ and thus the dimension is $3$.

For point 3 let consider $W'$ of $cx$.