direct sum vector spaces

Question

Looking at an exam past paper in preparation for my exam but unfortunately there are no solutions. The question involves a subspace, $U$, of $V= P_2$ which is spanned by $f(x)=x^2 + x + 1$. How would I find a subspace, say $W$, such that $V = U \oplus W$? I am struggling with how to formulate $W$ as I understand it cannot intersect with $U$?

Answer 1

$\dim P_{2}=3$. Since $\dim U=1$, it must be that $\dim W=2$. Try to convince yourself that something like $W=\text{span}\{1,x\}$ might work.