Finding constants of a polynomial in a real vector space

Question

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.

Answer 1

Taylor immediately kills this one.

But you can also show it with linear algebra. Let $V$ be the four-dimensional vector space of polynomials of degree $\leq 3$. The 4 maps $V \to \mathbb R, f \mapsto f^{(n)}(1)$ for $n=0,1,2,3$ are linear independent, hence they span the whole dual space. In particular the evaluation at $2$ is a linear combination of them.