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For applications to quasi-exactly solvable Schr\"odinger equations in quantummechanics, we establish the general conditions that have to be satisfied by thecoefficients of a second-order differential equation with at most $k+1$singular points in order that this equation has particular solutions which are$n$th-degree polynomials. In a first approach, we extend the Nikiforov-Uvarovmethod, which was devised to deal with hypergeometric-type equations (i.e., for$k=2$), and show that the conditions involve $k-2$ integration constants. In asecond approach, we consider the functional Bethe ansatz method in its mostgeneral form. Comparing the two approaches, we prove that under the assumptionthat the roots of the polynomial solutions are real and distinct, the $k-2$integration constants of the extended Nikiforov-Uvarov method can be expressedas linear combinations of monomial symmetric polynomials in those roots,corresponding to partitions into no more than two parts.

Extended Nikiforov-Uvarov method, roots of polynomial solutions, and functional Bethe ansatz method