Unified Inversion Method for Solving Polynomial Equations: A Reverse Detour to the Common Procedure

There are many solutions to polynomial equations that have been developed by mathematicians over the centuries. These methods adopt different approaches such as substitution, complex number algebra, trigonometry, reduction to depressed form, elimination, and decomposition of the original polynomial into solvable products of polynomials of lesser degree.  In this paper, a historical preview of the methods used to solve polynomial equations is provided, together with a review of recent methods demonstrated for solving polynomial equations. This paper also proposes a new unified method of solving polynomial equations based on the inversion of the nth roots of variables that will explicitly determine the root. The method is applicable to all polynomials within the limits of solvability of polynomials by radicals. The method follows a reverse route to the common methods and logically finds roots that are algebraically expressed as radicals of real numbers, although the formulation of the solution starts with inversion by finding the nth root of either real or complex numbers. By contrast, methods such as Cardan’s solution to cubic equations give solutions that have cube roots of complex numbers, whereas the roots are real numbers. The proposed method is simple and intuitive to understand and use. Examples have been provided to demonstrate the application of the proposed method.