孟飞飞的毕业论文

学校代码:11059学号:1107011032HefeiUniversity毕业论文（设计）BACHELORDISSERTATION论文题目:带余除法及其应用研究学位类别:理学学士学科专业:信息与计算科学作者姓名:孟飞飞导师姓名:余海峰完成时间:2015年05月03日带余除法及其应用研究摘要本文的主旨思想是带余除法的简单介绍以及带余除法在日常生活中的应用，整片论文都围绕带余除法来展开论述，先是介绍带余除法的来源及课题意义，然后通过整数的带余除法和多项式的带余除法让大家对带余除法的应用有一个更深的认识。最后通过实例来展现其在应用研究中所起到的作用。本文的正文是介绍整数和多项式的带余除法，从这二个层面可以认识到带余除法是一种普遍应用于生活中的思想。可以这样说多项式的带余除法是整数带余除法的推广，所以有必要对整数带余除法进行介绍，多项式的带余除法中将涉及辗转相除法的介绍，整除的基本概念与基本性质、最大公因式、公共根、重根以及一元多项式矩阵的相关性质。下面就开始进入本文的正题吧!关键词：一元多项式带余除法辗转相除法最大公因式一元多项式矩阵WithmorethandivisionanditsapplicationresearchabstractPurposeofthisarticleismorethanwithdivisionofsimpleintroduction,andwiththeapplicationofthedivisioninthedailylife,thewholepieceofpaperaroundwithyutodiscourseuponthedivision,firstintroducedmorethanwiththesourceofthedivisionandthetopicsignificance,andthenthroughmorethanmorethanwithdivisionandpolynomialwithintegerdivisionleteverybodytotakeovertheapplicationofthedivisionhasadeeperunderstanding.Finallybyanexampletoshowitsapplicationinplayaroleinthestudy.Thebodyofthispaperistointroducemorethanintegerandpolynomialdivision,fromthetwoaspectscanberealizedwithresidualdivisionisacommonusedinthelifeofthought.Morethancansaythispolynomialwithdivisionisthedevelopmentofmorethanintegerwithdivision,soitisnecessarytocarryoutmorethanintegerwithdivision,polynomialwithresidualdivisionwillinvolvedivisionalgorithmisintroduced,thebasicconceptofdivisibleandbasicproperties,thebiggestcommonfactor,publicroot,rootandthecharactersofoneyuanpolynomialmatrix.Thefollowingbegantogetintothisbusiness!Keywords:morethanoneyuanpolynomialdivisionDivisionalgorithmThegreatestcommonfactorisoneyuanpolynomialmatrix目录第一章前言...................................................................................................................................51.1研究背景...........................................................................................................................51.2课题意义...........................................................................................................................7第二章整数的带余除法...............................................................................................................82.1整数带余除法的解释及证明...........................................................................................82.2整数带余除法的一些性质...............................................................................................82.3最大公约数与辗转相除法................................................................................................82.4整除的进一步性质与最小公倍数...................................................................................9第三章多项式的带余除法.....................................................................................................113.1多项式带余除法的定理及其证明..................................................................................113.2带余除法的二种计算格式.............................................................................................123.2.1普通除法（长除法）.........................................................................................123.2.2竖式除法.............................................................................................................123.2.3综合除法.............................................................................................................13第四章带余除法在解题中的应用.............................................................................................144.1有关两个多项式除法与整除关系问题.........................................................................144.2辗转相除法是带余除法的特殊应用.............................................................................154.2.1辗转相除法计算两个多项式的最大公因式及它们与最大公因式的组合关系.........................................................................................................................................154.2.2求两个多项式的公共根.....................................................................................164.2.3解有关多项式的重根，重因式问题.................................................................174.3求函数值f(a)................................................................................................................174.4解有关有理数域上的因式分解及有理根.....................................................................184.5带余除法在矩阵多项式中的应用.................................................................................194,5.1关于矩阵多形式可逆的判定.............................................................................194.5.2有关矩阵最小多项式的问题.............................................................................20参考文献.........................................................................................................................................22致谢............................................................................................................................................23第一章前言1.1研究背景带余除法（也称为欧几里德除法）是数学中的一种基本算术计算方式。给定一个被除数a和一个除数b，带余除法给出一个整数q和一个介于一定范围的余数r，使得该等式成立：a=bq+r。一般限定余数的范围在0与b之间，也有限定在-b/2与b/2之间。这样的限定都是为了使得满足等式的q有且仅有一个。这时候的q称为带余除法的商。带余除法一般表示为：a/b=q…r。表达为：“a除以b等于q，余r”。最常见的带余除法是整数与整数的带余除法（被除数a和除数b都是整数），但实数与整数乃至实数与实数的带余除法也有应用。对一般的抽象代数系统，能够进行带余除法的都是具有欧几里德性质的系统。如果余数为零，则称b整除a。一般约定