A Critical Review of Chapter 1: Factor Equations and Factor Symmetry Theorem

This paper provides a comprehensive review and critical analysis of Chapter 1 of the book Factor Equation Symmetry Theorem and Its Applications by Meng Qinghua (Huilian Publishing House, 2025). The book proposes a novel framework for solving algebraic equations, termed “Factor Equations,” and introduces the “Factor Symmetry Theorem” as a foundational tool for finding rational solutions to univariate polynomial equations and potentially addressing classical Diophantine problems. Chapter 1 establishes the theoretical groundwork, defining factor equations, revising the Fundamental Theorem of Arithmetic to include the factor ‘1’ as a symmetric element, and presenting a method for determining roots based on factor structure component analysis. This review examines the mathematical rigor of these definitions, the logic of the symmetry theorems, and the practical efficacy of the proposed root-finding methods compared to established techniques like the Rational Root Theorem. While the author’s perspective offers an unconventional algebraic intuition, this review critically evaluates the necessity and novelty of the introduced concepts.