Let $Γ$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(λ)\in \mathbb{C}[λ]$, of degree $4$ depending only on the intersection numbers of $Γ$ and such that $T(η)\geq 0$ holds for any non-principal eigenvalue $η$ of the local graph $Γ(x)$ for any vertex $x\in V(Γ)$. We call $T(λ)$ the Terwilliger polynomial of $Γ$. In this paper, we give an explicit formula for $T(λ)$ in terms of the intersection numbers of $Γ$ and its dual eigenvalues. We then apply this polynomial to show that all pseudo-partition graphs with diameter at least $3$ are known.