Solving algebraic equations using Cardano-type formulas

E. S. Balko^a, N. K. Norets<sup>b

a</sup> Tomsk State University
^b V. I. Vernadsky Crimean Federal University, Simferopol

Abstract: The purpose of the project is to derive formulas for the roots of classes of solvable polynomials of higher degrees according to the principle of derivation of the Cardano formula. To achieve this goal the tasks were set:
– To familiarize yourself with the procedure of derivation of the Cardano formula for cubic equations and with the study of this formula depending on the sign of the discriminant.
– Define the classes of equations of higher degrees that are solvable in radicals.
– Derive a Cardano-type formula for a selected class of equations of the 5th degree and higher and analize it depending on the sign of the discriminant.
– Consider the examples.
The object of the research work are classes of algebraic equations of higher degrees, for which it is possible to derive a Cardano-type formula. The subject of the investigation is the type of roots of the algebraic equation of the selected classes depending on the sign of the discriminant. The methods of analogy and the analytical method of solving equations were used. The hypothesis is that the derivation of Cardano-type formulas for a selected class of algebraic equations of higher degrees is possible and that the obtained roots of this equation have a structure similar to the roots of a cubic equation.
After completion of this work, the following results can be noted.
The general form of the algebraic equation of degree $2k+1$, ($k$ an integer) is presented, for which Cardano-type formulas for calculation roots can be derived.
The root formulas for a certain class of algebraic equation are derived similarly to the derivation of the Cardano formula.
Solvable equations of the 5th, 7th, 9th and 11th degrees are examined in detail.
The similarity of the type and structure of the roots of the cubic equation and equations of a certain class of higher degrees, their belonging to a set of numbers depending on the sign of the discriminant, are noted.
The aim of the investigation has been achieved. he tasks have been solved. The hypothesis out to be true.

Keywords: Cardano's formulas, Cardano-type formula, equation, solvable polynomials, fifth degree, higher degrees, roots.

UDC: 512.1

MSC: 34A09