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

The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to the pseudo-partition graphs