Question: Prove that there are real numbers $a_1,a_2,a_3,a_4$ such that for any real polynomial $p$ of degree $ \leq 3$, $p(2)=a_1p(1)+a_2p'(1)+a_3p''(1)+a_4p'''(1)$.
I'm not really sure where to start. I sense one would have to denote $V$ as the real vector space of polynomials of degree $\leq 3$ and then find a basis for $V^*$ (the dual space), but I'm not sure how to do this. What would be a strategy for solving a problem like this?
This problem is from a linear algebra course.