多项式理论在初等数学中的应用——高次方程组与不等式的求解方法探究

Applications of Polynomial Theory in Elementary Mathematics—An Exploration of Methods for Solving Higher-Degree Systems of Equations and Inequalities

摘要: 多项式理论是代数学的核心基础，其方法在初等数学问题的求解中具有广泛应用。本文以多项式整除、辗转相除法及对称多项式等核心理论为依托，探究其在初等数学高次方程组求解与不等式证明中的具体应用。通过典型例题的解析，展现多项式方法化繁为简、直击问题本质的解题优势，为高次方程组与不等式的求解构建系统化思路，同时直观体现高等代数与初等数学的深度衔接及内在关联。

Abstract: Polynomial theory serves as the fundamental cornerstone of algebra, with its methodologies widely applied in solving elementary mathematics problems. Building upon core concepts such as polynomial divisibility, method of successive division, and symmetric polynomials, this study explores their practical applications in solving higher-order systems of equations and proving inequalities in elementary mathematics. Through detailed analysis of representative examples, the research demonstrates the advantages of polynomial methods in simplifying complex problems and addressing core mathematical issues. It establishes a systematic approach for solving higher-order equations and inequalities while visually illustrating the profound connections and intrinsic relationships between advanced algebra and elementary mathematics.