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The General Quintic Equation, its Solution by Factorization into Cubic and Quadratic Factors

Samuel Bonaya Buya

I present a method of solving the general quintic equation by factorizing into auxiliary quadratic and cubic equations. The aim of this research is to contribute further to the knowledge of quintic equations. The quest for a formula for the quintic equation has preoccupied mathematicians for many centuries.

The monic general equation has six parameters. The objectives of this presentation is to, one, seek a factorized form of the general quintic equation with two exogenous and four indigenous parameters, two, to express the two exogenous parameters of factorization as a function of the original parameters of the general quintic equation.

In the process of factorization two solvable simultaneous polynomial equations containing two exogenous and four original parameters are formed. Each of the exogenous parameters is related to the coefficients of the general quintic equation before proceeding to solve the auxiliary quadratic and cubic factors.

The success in obtaining a general solution by the proposed method implies that the Tschirnhausen transformation is not needed in the search for radical solution of higher degree general polynomial equations.

As a way forward factorized form for solution of the general sextic and septic equations will be presented to pave way for their algebraic solution.