Polynomial Sub-Spaces

Question

Let $ V =\mathbb{R}_2[X]$ and $U = \{p\in V \mid p(-1) = p(2)\}$

Show that $U$ is a Linear Subspace of $V$ , and find a basis for $U$.

Complete the basis of $U$ to a basis of $V$.

I understand that all I need to do for the first part is find the basis, because every $p\in V$ . I am struggling with the polynomial format , and completing the basis.

Answer 1

Let $p(x)=ax^2+bx+c$. Then $p \in U \iff p(-1)=p(2) \iff b=-a.$

A Basis for $U$ is, for example, $\{1,x^2-x\}.$ Can you proceed ?