Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem

Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.