<abstract xmlns="http://www.w3.org/1999/xhtml">The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, <italic>u</italic>2<italic>− g</italic>2 = (<italic>u</italic> + <italic>g</italic>)(<italic>u − g</italic>). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.</abstract>

The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u2− g2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

Insert a Root to Extract a Root of Quintic Quickly

Department of Electronics & Communication Engineering

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Insert a Root to Extract a Root of Quintic Quickly"}

The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up,...

Insert a Root to Extract a Root of Quintic Quickly

Published Online: Jul 18, 2019

Page range: 153 - 158

Received: Feb 18, 2018

Accepted: Dec 08, 2018

DOI: https://doi.org/10.2478/amsil-2018-0013

Keywords
solvable quintic equation, principal quintic equation, sextic equation

© 2019 Raghavendra G. Kulkarni, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Insert a Root to Extract a Root of Quintic Quickly

Published Online: Jul 18, 2019

Page range: 153 - 158

Received: Feb 18, 2018

Accepted: Dec 08, 2018

DOI: https://doi.org/10.2478/amsil-2018-0013

Keywordssolvable quintic equation, principal quintic equation, sextic equation

© 2019 Raghavendra G. Kulkarni, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
solvable quintic equation, principal quintic equation, sextic equation