关于一元多项式的因式分解.rar

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摘要  在高等代数中, 常常会遇到一元多项式的因式分解问题, 例如, 含字母系数的方程组求解、矩阵的最小多项式的求解、判断线性变换及矩阵是否可以对角化等. 虽然我们有因式分解及唯

一性定理, 即:数域中每一个次数≥1的多项式都可以唯一地分解成 中一些不可约多项式的乘积, 但一元多项式的因式分解没有具体公式可以利用, 因而, 有关这类问题的求解一直以来是学习的一个难点. 在本文中,我们对一元多项式的因式分解的方法进行了初步的探索, 用实例来比较综合除法,待定系数法,辗转相除法等七种方法的优劣。

关键词 一元多项式; 因式分解; 综合除法; 待定系数法

 

Abstract: In higher algebra, we often encounter problems of factorizing the polynomial of one variable, Such as solving the equations having letters as coefficients , solving the minimal polynomial of a matrix, determining whether a linear transformation or a matrix could be diagonalized, etc. Although, the unique factorization theorem holds, i.e. for each polynomial with degree more than or equal to 1 in F[x] can be uniquely factorized into a product of some irreducible polynomial, there is no method that can be applied to any polynomial for factorization. In this paper, we investigate some techniques of the factorization of the polynomial of one variable, and measures the pros and cons of some techniques by using them to factorizing some specific polynomials.

Keywords   the polynomial of one variable ; factorization; comprehensive  division; undetermined coefficients

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