A General Approach to Exact Polynomial Solutions: Demonstrations for Quintic and Sextic Equations and a Higher-Degree Conjecture

A novel decomposition of high-degree polynomials is presented in this paper by reformulating polynomial solving as a matrix decomposition problem, leveraging linear algebra to retrieve exact solutions whereas current methods rely on approximations. This approach is demonstrated on subfamilies of quintic and sextic polynomials, supported by numerical examples. The full matrix decompositions for each subfamily are provided in the appendix, from which the solutions naturally arise. This framework also suggests a conjecture for polynomials of degree nine and higher, thus outlining conditions for exact solvability and inviting further exploration.