一无二次方程的整数根的求法

一元二次方程ax2+bx+c=0(a≠0)是初中数学的重点内容,本文介绍解整系数一元二次方程含有参数的整数根问题的五种基本方法. 1.求根公式法若能将方程ax2+bx+c=0的根表示成有理式则可结合整除性求解. 例1 求整数m,使关于x的一元二次     

三次方程根的情况的简明结论

由“实系数一元n(∈N*)次方程的虚根成对出现”知,实系数一元三次方程的根有且只有四种情形:(1)有三个不同实根;(2)有一个二重实根和一个实根;(3)有一个三重实根;(4)有一个实根和两个共轭虚根.本文用导数研究它们在何时出现.先看首项系数是正数的一元三次方程f(x)=ax3 bx2 cx d     

一元多项式的根

对于1’>O,如果了(劣)~‘扩+‘一:扩一’十…十al劣+a0是复系数一元:次多项式,那么方程f(:)二。,即 a·‘,+‘一、‘,一‘+…+a,:+a。=0②①叫做复系数一元二次方程.方程②的根,也称多项式① 的根. 一27一中学数学(湖北)1992.12 类似地,如果j(,)是实系数(或有理系数、整系数等)一元:(、>0)次多项式,那么方程j(幻二O叫做实系数(或有理系数、整系数等)一元:次方程. 关于多项式的根的个数有以下重要定理: l代数墓本定理一元:次多项式在复数集中至少有一个根. :799年伟大的数学家高斯证明了这一重要定理· 2根的个数定理一元二次多项式有且仅有…     

巧用根与系数的关系解竞赛题

根与系数的关系是一元"次方程(n∈N~*)的重要性质,本文通过实例来说明巧用一元二次(三次)方程的根与系数的关系解竞赛题.1.利用一元二次方程根与系数的关系解题当已知条件中出现或者通过转化后出现两数之和、两数之积时,可考虑利用根与系数的关系来构造一元二次方程(或函数)来解题.     

含参数的一元三次方程问题

含参数的一元三次方程问题题型较多,涉及知识面广,求解难度较大,因而在数学竞赛中时有出现.本文将其解法作一介绍.含参数的一元三次方程问题概括地说有三大类:一类是由方程根的性质确定参数,另一类是由参数的性质确定方程的根,再一类是由方程的根和参数的性质确定未知,分述如下.1由方程根的性质确定参数即已知一元三次方程的根具有某些性质,求参数的值、取值范围及参数之间的关系等.例1(1995年全国高中数学联赛)求一切实数p,使得三次方程5x3-5(p 1)x (71p-1)x-66p 1=0(1)有三个根均为正整数.解观察可知x=1为(1)的一个根.由综合除法可得5x2-5…     

分式线性函数的亚纯迭代根

迭代根问题是嵌入流的一个弱问题.关于单调函数的迭代根已有较多结论.但是对非单调函数迭代根的研究却很困难的.分式线性函数是一类实数域上的非单调函数．本文对复平面上分式线性函数的迭代根进行了研究.将分式线性函数的迭代函数方程与一个商空间上的矩阵方程对应,并运用一个求解矩阵根的方法,得到其所有亚纯迭代根的一般公式.并且确定了不同情形下分式线性函数迭代根的准确数目. 作为应用,分别给出了函数$z$和函数$1/z$全部亚纯迭代根.     

求方程近似根的双切线法

<正> 在实际工作中常常要求出某一方程具有足够准确度的根,但只有一次、二次代数方程及一些特殊类型的方程才有准确解法,一般都要用近似解法。若求方程f(x)=0……(1)的近似根,我们知道可用切线法求近似根,而切线法的计算公式是x_(n+1)=x_n+h_(n+1)……(·),其中矫正值h_(n+1)=-(f(x_n)/f′(x_n)),这里x_n为     

数列问题中的特征根方法与拆项方法

本文介绍循环数列、某些分式递推式确定的数列及阶差数列,并利用特征根方法或拆项方法求其通项或前n项的和。 一循环数列 若数列{。}的项满足 a.=A:么_.+凡‘一2+…+儿山_、月>毛川.lJ称{司为k阶循环数列,这里乏是固定的正整数,Al,九,…,儿是与n无关的常数,A,手0.(l)式称为1‘}的循环方程,方程 犷=A,犷一,+九xx一“+…+A.(2)称为1.}的特征方程;(2)的根称为1司的特征根.不难证明,等差数列是二阶循环数列,而等比数列是一阶循环数列。 显然,满足同一个k阶循环方程的数列有无限多个。为了确定一个数列,还需知道数列的某h个项的具体值。常见的…     

实系数一元四次方程的矩阵解法

1　引言许多实际问题 ,尤其是方阵的特征值与某些微分方程的求解往往归结为特征方程———一元ｎ次方程根的求解问题 .然而 ,当方程的次数大于或等于四次时其一般解的获得就不那么容易了 .众所周知 ,一元三次方程有求根公式———卡尔丹公式 ,而一元四次方程就没有确切的求根公式 .本文旨在给出一种通过矩阵变换来求一元四次方程根的新方法 .2　引理不失一般性 ,设实系数一元四次方程为 :ａ0 ｘ4+ａ1 ｘ3+ａ2 ｘ2 +ａ3ｘ +ａ4=0 (1 )(ａ0 ≠ 0 ,ａｉ ∈Ｒ ,0 ≤ｉ≤ 4)引理 1　记ＹＴ =(ｘ2 ,ｘ ,1 ) ,Ａ=ａ0ａ1 2 ｕａ1 2 ａ2 - 2ｕ ａ32ｕ…     

问题与解答

一本期问题 1若x、万满足椭圆方程x“/1984十g活=1985,求证.x一卜岁.‘1985. 2求证‘鉴’4*,能被1 05。,整除,商为1 985汤。 3设x、夕)0,且x+,=1 986,求别‘’‘’的最大值。 南昌华东交大附中叶柯提供 4设无为正整数,且一元二次万程(无一1)x生一Px十k二o有两个正整数根,试求k“P(pp+无“)+l的值。 5第四世纪,有个希腊数学家在他所著的算术书中,有一道这样的问题: x‘一60为一个完全平方数,求x,你能不能解答? 6已知x、,、:为三个正实数,且x一卜y十z=3,1/x+1/,+r/z=3,求x“,忿+:,的值 皖祈门芦溪木材收购知长进球提供 7如果二、n、P是方程x…     

代数方程在任意圆外根的个数

从复变函数论知,复多项式f(u)=sum from l=0 to n (a_1u~(n-1))在u=u_0点的Taylor展开式是u的恒等式:     

Maslov dequantization and the homotopy method for solving systems of nonlinear algebraic equations

The Maslov dequantization allows one to interpret the classical Gräffe-Lobachevski method for calculating the roots of polynomials in dimension one as a homotopy procedure for solving a certain system of tropical equations. As an extension of this analogy to systems of n algebraic equations in dimension n, we introduce a tropical system of equations whose solution defines the structure and initial iterations of the homotopy method for calculating all complex roots of a given algebraic system. This method combines the completeness and the rigor of the algebraicgeometrical analysis of roots with the simplicity and the convenience of its implementation, which is typical of local numerical algorithms.     

An interval iteration for multiple roots of transcendental equations

An iteration method for roots of algebraic functions with roots of multiplicity greater than one is established using tools and techniques from interval arithmetic. The method is based on an interval iteration functions for multiple roots and it retains the convergence order of the underlying iteration method while preserving global convergence over an initial interval. A number of simple examples are provided to show that the method is feasible and that it produces reasonable results.     

Algebraic polynomials and moments of stochastic integrals

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.     

Solving triangular algebraic systems by means of simultaneous iterations

Our aim in this paper is to extend a variant of the Weierstrass method for the simultaneous computation of the solutions of a triangular algebraic system of equations. The appropriate tools are the symmetric functions of the roots of a polynomial. Using these symmetric functions we give another equivalent formulation for the search of all the roots of a triangular algebraic system. Using the latter formulation our method consists in solving a more simple system (where partial degrees of all the equations do not exceed 1) by Newton’s method. The quadratic convergence of our method is an immediate consequence of Newton’s method and need not be proved explicitly. This revised version was published online in June 2006 with corrections to the Cover Date.     

ON THE METHODS FOR FINDING ROOTS OF ALGEBRAIC EQUATIONS WITH WEIERSTRASS'''' CORRECTIONS

This is a study of the Durand-Kerner and Nourein methods for finding the roots of a given algebraic equation simultaneously. We consider the conditions under which the iterative methods fail. The numerical example is presented.     

On One Method for Factorization of Algebraic Polynomials

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.     

An approach to solving nonlinear algebraic systems. 2

New methods of solving nonlinear algebraic systems in two variables are suggested, which make it possible to find all zero-dimensional roots without knowing initial approximations. The first method reduces the solution of nonlinear algebraic systems to eigenvalue problems for a polynomial matrix pencil. The second method is based on the rank factorization of a two-parameter polynomial matrix, allowing, us to compute the GCD of a set of polynomials and all zero-dimensional roots of the GCD. Bibliography: 10 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 71–96     

Numerical solution of algebraic fuzzy equations with crisp variable by Gauss–Newton method

In this paper we introduce an algebraic fuzzy equation of degree n with fuzzy coefficients and crisp variable, and we present an iterative method to find the real roots of such equations, numerically. We present an algorithm to generate a sequence that can be converged to the root of an algebraic fuzzy equation.     

Über die Verteilung von primen primitiven Wurzeln in algebraischen Zahlkörpern

Generalizing a method introduced by Elliott in the rational case to number fields in an appropriate way, asymptotic estimates are given for the number of algebraic primes in certain parallelotopes which are primitive roots for almost all (in a certain sense) prime ideal moduli. The proofs depend upon a fundamental lemma of Selberg's rational sieve method and make use of the large sieve in the setting of an algebraic number field.