Matrix approach to solve polynomial equations

Abstract

Polynomials are widely employed to represent numbers derived from mathematical operations in nearly all areas of mathematics. The ability to factor polynomials entirely into linear components allows for a wide range of problem simplifications. This paper presents and demonstrates a novel straightforward approach to solving polynomial problems by converting them to matrix equations. Each polynomial of degree n can be decomposed into a sum of degree ⌈ n 2 ⌉ polynomials squared, i.e., ∑ i = 0 n a i x i = ∑ i = 1 ⌈ n 2 ⌉ + 1 ∑ j = 0 j = ⌈ n 2 ⌉ b i , j x j 2 . It follows that the complexity of factorizing a polynomial of degree 2 n is equivalent to that of factorizing polynomial of degree 2 n − 1 . The proposed method for solving fourth-degree polynomials will be a valuable contribution to linear algebra due to its simplicity compared to the current method. This work presents a unique approach to solving polynomials of four or fewer degrees and presents new possibilities for tackling larger degrees. Additionally, our methodology can also be used for educational purposes.

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