Difficult# Finite Dimension: Polynomial Subspace and Second Derivatives

LINALG-RIJXUD

Let $\mathcal{P}_n$ denote the set of all real-coefficient polynomials of degree at most $n$. This is a finite-dimensional vector space. Let $W = \{f(x) \in \mathcal{P}_{n}\mid f''(x) = 0\}$, which is a subspace of $\mathcal{P}_{n}$ for $n\ge 2$. What is $\dim(W)$?