A closed solution to a special polynomial trinomial equation and semi-analytical roots for a general algebraic equation

Abstract

We suggest a closed solution for the roots of polynomial trinomial algebraic equation $$z^n+xz^{n-1}-1=0$$ with an appropriate $x$. This solution is a minor modification to the work of Mikhalkin (Mikhalkin E N, 2006. On solving general algebraic equations by integrals of elementary functions, Siberian Mathematical Jounral, 47(2), 301-306). This modification, together with Mikhalkin's integral formula, provides a relatively simple analytical expression for the solution to a general algebraic equation when the polynomial coefficients are over the corresponding convergent domain. Numerical examples show that this expression can be another alternative to finding numerically the roots of a general polynomial algebraic equation when the integral involved exists and is calculated correctly.