On the Exact Solvability of Polynomial Families: Quintic and Sextic Cases with a Conjectural Extension to Higher Degrees

Abstract

We develop a framework for identifying conditionally solvable families of quintic and sextic polynomials by reformulating polynomial solving with the coefficient relations as a constrained matrix decomposition problem. By enforcing rank-2 conditions on an associated symmetric matrix, the original polynomial equation can, for certain structured families, be reduced to lower-degree components. Since polynomial equations of degree at most four are solvable by radicals, such reductions provide explicit radical solutions whenever the reduction succeeds. We emphasize that the rank-2 condition is not sufficient for solvability in general; rather, it identifies special coefficient families for which the resulting reduced equations fall within solvable classes. The parameters introduced in the construction arise from solving an auxiliary algebraic system obtained via Gaussian elimination, and different parameter choices correspond to distinct solvable subfamilies. Several quintic and sextic examples are presented to illustrate the method. Finally, we propose a conjectural extension to higher-degree polynomials for degrees nine and above, formulated in a manner consistent with classical Galois theory, thus outlining conditions for exact solvability and inviting further exploration.