An Alternative Approach to Solving Cubic and Quartic Polynomial Equations

Aims: The aim of the research study was to develop a more direct and intuitive approach for the solution of polynomial equations of degree 3 and four. Study Design:  The study employed equivalent polynomial substitution that is more intuitive and direct to formulate than the traditional formulations and one that is easily solvable. Place and Duration of Study: The study has been undertaken by the author at the university of Eswatini in the period from February to March 2021. Methodology: Two alternative procedures have been presented for the analytical solution of cubic and quartic equations and demonstrated with worked examples. The solution is derived through a direct procedure without involving intermediate variable substitution. Results: For cubic equations, the solution provides explicit expression of an equivalent cubic that is formed directly in terms of the original variable x. As such, the formula is intuitive and simple to derive or understand as well as apply. For the quartic equations, the same decomposition form is used as that of the cubic equation using two quadratic polynomials that have symmetric form thus making it easy to develop the solution as well as solve the equations Conclusion: The alternative formula is easy to formulate and solve and provides a more intuitive basis for understanding and solving polynomial equations.