Beyond the Quadratic Equations and the N-D Newton-Raphson Method

Abstract

Abstract: The objective of this technical paper was to propose an approach for solving polynomials of degree higher than two. The main concepts were the decomposition of a polynomial of a higher degree to the product of two polynomials of lower degrees and the n-D Newton-Raphson method for a system of nonlinear equations. The coefficient of each term in an original polynomial of order m will be equated to the corresponding term from the collected-expanded product of the two polynomials of the lower degrees based on the concept of undetermined coefficients. Consequently a system of nonlinear equations was formed. Then the unknown coefficients of the decomposed polynomial of the lower degree of the two decomposed polynomials would be eliminated from the system of the nonlinear equations. After that the unknown coefficients in the decomposed polynomial of the higher degree would be obtained by the n-D Newton-Raphson. Finally the unknown coefficients for the decomposed polynomial of the lower degree would be obtained by back substitutions. In this technical paper the formulations for the decomposed polynomials would be derived for the polynomials degree three to nine. Several numerical examples were also given to verify the applicability of the proposed approach.