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2019 | 33 | 153--158
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Insert a Root to Extract a Root of Quintic Quickly

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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, $u^2 - g^2 = (u + g)(u - g)$. 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.(original abstract)
Opis fizyczny
  • PES University Department of Electronics & Communication Engineering
  • Adamchik V.S., Jeffrey D.J., Polynomial transformations of Tschirnhaus, Bring and Jerrard, ACM SIGSAM Bull. 37 (2003), no. 3, 90-94.
  • Bewersdorff J., Galois Theory for Beginners. A Historical Perspective. Translated from the second German (2004) edition by David Kramer, American Mathematical Society, Providence, 2006.
  • 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.
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