The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

Ilija Tanackov, Ivan Pavkov and Željko Stević
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Ilija Tanackov: Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia
Ivan Pavkov: Faculty of Mathematics and Computer Science, Alfa BK University, Palmira Toljatija 3, 11000 Belgrade, Serbia
Željko Stević: Faculty of Transport and Traffic Engineering Doboj, University of East Sarajevo, Vojvode Mišića 52, 74000 Doboj, Bosnia and Herzegovina

Mathematics, 2020, vol. 8, issue 5, 1-18

Abstract: An arbitrary univariate polynomial of n th degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of n th degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an n th-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of n th degree ( n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

Keywords: nonlinear approximate method; quintic equation; Pentanacci sequence; Abel’s impossibility theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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