AM  Vol.11 No.2 , February 2020
Solution to Polynomial Equations, a New Approach
Abstract: A new approach for solving polynomial equations is presented in this study. Two techniques for solving quartic equations are described that are based on a new method which was recently developed for solving cubic equations. Higher order polynomial equations are solved by using a new and efficient algorithmic technique. The proposed methods rely on initially identifying the vicinities of the roots and do not require the use of complicated formulas, roots of complex numbers, or application of graphs. It is proposed that under the stated conditions, the methods presented provide efficient techniques to find the roots of polynomial equations.
Cite this paper: Tehrani, F. (2020) Solution to Polynomial Equations, a New Approach. Applied Mathematics, 11, 53-66. doi: 10.4236/am.2020.112006.

[1]   Atkinson, K.E. (1989) An Introduction to Numerical Analysis. John Wiley & Sons, Inc., Hoboken.

[2]   Scavo, T.R. and Thoo, J.B. (1995) On the Geometry of Halley’s Method. The American Mathematical Monthly, 102, 417-426.

[3]   Akritas, A.G., Strzebonski, A.W. and Vigklas, P.S. (2008) Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots. Nonlinear Analysis: Modelling and Control, 13, 265-279.

[4]   Collins, G.E. (2001) Polynomial Minimum Root Separation. Journal of Symbolic Computation, 32, 467-473.

[5]   Edelman, A. and Kostlan, E. (1995) How Many Zeros of a Random Polynomial Are Real? Bulletin of the American Mathematical Society, 32, 1-37.

[6]   Hirst, H.P. and Macey, W.T. (1997) Bounding the Roots of Polynomials. The College Mathematics Journal, 28, 292-295.

[7]   Sun, Y.J. and Hsieh, J.G. (1996) A Note on the Circular Bound of Polynomial Zeros. IEEE Transactions on Circuits and Systems, 43, 476-478.

[8]   Dalal, A. and Govil, N.K. (2017) On Comparison of Annuli Containing All the Zeros of a Polynomial. Applicable Analysis and Discrete Mathematics, 11, 232-241.

[9]   Gao, L. and Govil, N.K. (2018) Annular Bounds for the Zeros of a Polynomial. International Journal of Mathematics and Mathematical Sciences, 2018, Article ID: 6047387.

[10]   Tehrani, F.T. (2016) A Simple Approach to Solving Cubic Equations. The Mathematical Gazette, 100, 225-232.

[11]   Aude, H.T.R. (1949) Notes on Quartic Curves. The American Mathematical Monthly, 56, 165-170.

[12]   Faucette, W.M. (1996) A Geometric Interpretation of the Solution of the General Quartic Polynomial. The American Mathematical Monthly, 103, 51-57.

[13]   Kappe, L.C. and Warren, B. (1989) An Elementary Test for the Galois Group of a Quartic Polynomial. The American Mathematical Monthly, 96, 133-137.

[14]   Rees, E.L. (1922) Graphical Discussion of the Roots of a Quartic Equation. The American Mathematical Monthly, 29, 51-55.

[15]   Shmakov, S.L. (2011) A Universal Method of Solving Quartic Equations. International Journal of Pure and Applied Mathematics, 71, 251-259.

[16]   Froberg, C.E. (1965) Introduction to Numerical Analysis. Addison-Wesley Publishing Company Inc., Reading.

[17]   Thomas, J.M. (1941) Sturm’s Theorem for Multiple Roots. National Mathematics Magazine, 15, 391-394.

[18]   Shamseddine, K. and Berz, M. (2010) Analysis of the Levi-Civita Field, a Brief Overview. In: Berz, M. and Shamseddine, K., Eds., Contemporary Mathematics, Advances in p-Adic and Non-Archimedean Analysis, Vol. 508, The American Mathematical Society, Providence, 215-237.