Insert a root to extract a root of quintic quickly
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 oddpowers 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.
Keywords
solvable quintic equation; principal quintic equation; sextic equation
References
2. Bewersdorff J., Galois Theory for Beginners. A Historical Perspective. Translated from the second German (2004) edition by David Kramer, American Mathematical Society, Providence, 2006.
3. von Tschirnhaus E.W., A method for removing all intermidiate terms from a given equation, Acta Eruditorum, May 1683, 204–207. Translated by R.F. Green in ACM SIGSAM Bull. 37 (2003), no. 1, 1–3.
Department of Electronics & Communication Engineering, PES University, India India
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