國高中因式分解方法論

因式分解為國中數學教材之一，是學生學習如何求方程式的解的先備知識，因此這個主題相當重要。學生對於較複雜的題目，有時會不知如何思考，或是不知該選用何種方法較合適而快速，因此本篇論文對於國高中因式分解教材，將單變數多項式因式分解與多變數多項式因式分解整理及探討。關於因式分解的方法論，除了運用基本乘法公式以外，還有提公因數或公因式、十字交乘法、雙十字交乘法、配方法、兩兩成對法、牛頓一次因式檢查法、長除法、五星法（三次多項式有理根因式分解法）、三次多項式無理根因式分解法、係數對稱法、因式定理與餘式定理、乘開再組合等。本篇論文收集各家版本的各種方法原理並舉例及詳列算式，期能讓學生清楚明白其中之奧妙。

關鍵字

因式；倍式；因式分解

並列摘要

Given that factorization, one of mathematics curriculum in junior high school, is the prior knowledge of solving equation. This topic is very important. Sometimes, students do not know how to think or how to choose a method that is suitable and rapid for a complex examination question. Therefore, this article sorts out and discusses factorization of univariate polynomial and multivariate polynomial in junior and senior high school mathematics. Regarding the methodology of factorization, we can apply mechanically basic multiplication formula. In addition, put forward common factor, cross-product method, method of completing the square, long division, factorization of cubic polynomial with rational root, factorization of cubic polynomial with irrational root, method of substitution to equation with symmetrical coefficient, factor theorem and Polynomial remainder theorem, multiply and regroup. This article is aiming at helping students clearly understand the underlying rationale of factorization by collecting various kinds of methods and principles of various versions.