The Solvability of Certain Sixth-Degree Polynomial Equations and the Radical Expressions of Their Roots Under Specific Conditions

This article deals with sixth-degree polynomial equations and, more specifically, describes the necessary conditions under which the solutions of these polynomials can be expressed in closed-form radicals. For this purpose, eight different cases of sixth-degree polynomial equations will be discussed in detail. It is concluded that if a specific relation exists between the constant term and the remaining coefficients of a sixth-degree polynomial equation, then its solutions can be obtained by formulas expressed as functions of the initial coefficients and ultimately represented in terms of radicals. Each case of the polynomial discussed in this article is presented in the “problem–proof” format.