MODERN METHODS FOR SOLVING ALGEBRAIC EQUATIONS

Solving algebraic equations remains a central problem in mathematics, with applications spanning engineering, physics, computer science, and applied sciences. This article reviews modern methods for solving algebraic equations, including classical analytical techniques, numerical iterative approaches, and computer-algebra-based symbolic methods. We examine the evolution from traditional radical-based solutions to contemporary computational algorithms such as Newton-Raphson iteration, Bairstow's method, Durand-Kerner (Weierstrass) method, and homotopy continuation techniques. Particular attention is given to methods for solving high-degree polynomial equations, systems of nonlinear algebraic equations, and equations lacking closed-form solutions. The comparative efficiency, convergence properties, and computational complexity of these methods are discussed, along with their implementation in modern computer algebra systems (CAS) such as Mathematica, MATLAB, and Maple.This review aims to provide researchers and practitioners with a structured overview of current methodologies and to identify promising directions for future research in computational algebra.