Exercise 12

Find a basis for each of these subspaces of the space $\mathcal{P}_{3}$ of cubic polynomials.

• The subspace of cubic polynomials $p(x)$ such that $p(7)=0$

• The subspace of polynomials $p(x)$ such that $p(7)=0$ and $p(5)=0$

• The subspace of polynomials $p(x)$ such that $p(7)=0$, $p(5)=0$, and $p(3)=0$

• The space of polynomials $p(x)$ such that $p(7)=0$, $p(5)=0$, $p(3)=0$, and $p(1)=0$