2,296
Views
0
CrossRef citations to date
0
Altmetric
Notes

The Midpoints Between Roots Reveal the Quartic Equation

Pages 258-262 | Received 28 Feb 2019, Accepted 06 Jul 2019, Published online: 24 Feb 2020

References

  • Auckly, D. (2007). Solving the quartic with a pencil. Amer. Math. Monthly. 114(1): 29–39. DOI: 10.1080/00029890.2007.11920389.
  • Clifford, J. H., Lachance, M. (2013). Quartic coincidences and the singular value decomposition. Math. Mag. 86(5): 340–349. DOI: 10.4169/math.mag.86.5.340.
  • Clifford, J. H., Lachance, M. (2018). A generalization of the Bôcher–Grace theorem. Rocky Mountain J. Math. 48(4): 1069–1076. DOI: 10.1216/RMJ-2018-48-4-1069.
  • Faucette, W. M. (1996). A geometric interpretation of the solution of the general quartic polynomial. Amer. Math. Monthly. 103(1): 51–57. DOI: 10.2307/2975214.
  • Glenn, O. E. (1907). Notes on the geometrical representation of the roots of equations. Amer. Math. Monthly. 14(10): 163–170. DOI: 10.1080/00029890.1907.11997385.
  • Graustein, W. C. (1928). A geometrical method for solving the biquadratic equation. Amer. Math. Monthly. 35(5): 236–238. DOI: 10.1080/00029890.1928.11986822.
  • Hofmann, H. (1888). La solution géométrique de l’équation du quatrième degré. Nouvelles Annales de Mathématiques 3e série. 7: 120–133.
  • Hoppe, E. (1874). Construction der reellen Wurzeln einer Gleichung vierten oder dritten Grades mittelst einer festen Parabel. Archiv der Mathematik und Physik. 56: 110–112.
  • Hoppe, E. (1883). Construction der imaginären Wurzeln einer Gleichung vierten oder dritten Grades mittelst einer festen Parabel. Archiv der Mathematik und Physik. 69: 216–218.
  • Irving, R. (2013). Beyond the Quadratic Formula. Washington, DC: Mathematical Association of America.
  • McNamee, J. M., Pan, V. Y. (2013). Low-degree polynomials. Stud. Comput. Math. 16: 527–556.
  • Neumark, S. (1965). Solution of Cubic and Quartic Equations. Oxford, UK: Pergamon Press.
  • Nickalls, R. W. D. (2009). The quartic equation: invariants and Euler’s solution revealed. Math. Gaz. 93(526): 66–75. DOI: 10.1017/S0025557200184190.
  • Nickalls, R. W. D. (2012). The quartic equation: alignment with an equivalent tetrahedron. Math. Gaz. 96(535): 49–55. DOI: 10.1017/S0025557200003958.
  • Northshield, S. (2013). Geometry of cubic polynomials. Math. Mag. 86(2): 136–143. DOI: 10.4169/math.mag.86.2.136.
  • Parish, J. L. (2006). On the derivative of a vertex polynomial. Forum Geom. 6: 285–288.
  • Running, T. R. (1943). Graphical solutions of cubic, quartic, and quintic. Amer. Math. Monthly. 50(3): 170–173. DOI: 10.1080/00029890.1943.11991345.
  • Shmakov, S. L. (2011). A universal method for solving quartic equations. Int. J. Pure Appl. Math. 71(2): 251–259.
  • Vieille, J. (1857). Sur la construction des racines de l’équation du quatrième degré par l’intersection d’une parabole et d’un cercle. Nouvelles Annales de Mathématiques 1er série. 16: 453–456.
  • Weisstein, E. (2019). Vieta’s Formulas—From MathWorld, A Wolfram Web Resource. mathworld.wolfram.com/VietasFormulas.html