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Rational Root
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation a_nx^n+a_x^+\cdots+a_0 = 0 with integer coefficients a_i\in\mathbb and a_0,a_n \neq 0. Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution written in lowest terms (that is, and are relatively prime), satisfies: * is an integer factor of the constant term , and * is an integer factor of the leading coefficient . The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is . Application The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Polynomial
In mathematics, a quadratic function of a single variable is a function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished and the terms ''quadratic function'' and ''quadratic polynomial'' are nearly synonymous and often abbreviated as ''quadratic''. The graph of a real single-variable quadratic function is a parabola. If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or ''roots'') of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve more than one variabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PlanetMath
PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org, Ltd". PlanetMath was started when the popular free online mathematics encyclopedia MathWorld was temporarily taken offline for 12 months by a court injunction as a result of the CRC Press lawsuit against the Wolfram Research company and its employee (and MathWorld's author) Eric Weisstein. Materials The main PlanetMath focus is on encyclopedia, encyclopedic entries. It formerly operated a self-hosted forum, but now encourages discussion via Gitter. An all-inclusive PlanetMath ''Free Encyclopedia'' book of 2,300 pages is available for the encyclopedia contents up to 2006 as a free download PDF file. Content development model PlanetMath implements a specific content creation system called ''authority model''. An author who starts a ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Root-finding
Finding the roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the development of mathematics. It involves determining either a numerical approximation or a closed-form expression of the roots of a univariate polynomial, i.e., determining approximate or closed form solutions of x in the equation a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n = 0 where a_i are either real or complex numbers. Efforts to understand and solve polynomial equations led to the development of important mathematical concepts, including irrational and complex numbers, as well as foundational structures in modern algebra such as fields, rings, and groups. Despite being historically important, finding the roots of higher degree polynomials no longer play a central role in mathematics and computational mathematics, with one major exception in computer algebra. Overview Closed-form formulas Closed-form formulas for polyno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein's Criterion
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. Criterion Suppose we have the following polynomial with integer coefficients: Q(x) = a_n x^n + a_ x^ + \cdots + a_1 x + a_0. If there exists a prime number such that the following three conditions all apply: * divid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Content (algebra)
In algebra, the content of a nonzero polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit (ring theory), unit of the ring (mathematics), ring of the coefficients (and the multiplication of the primitive part by the multiplicative inverse, inverse of the unit). A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial. Gauss's lemma (polynomials), Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Polynomial Roots
In mathematics, a univariate polynomial of degree with real or complex coefficients has complex ''roots'' (if counted with their multiplicities). They form a multiset of points in the complex plane, whose geometry can be deduced from the degree and the coefficients of the polynomial. Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which define a disk containing all roots, or lower bounds on the distance between two roots. Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree with real coefficients, which is less than 1+\frac 2\pi \ln (n) for sufficiently large. Notation In this article, a polynomial is always denoted : p(x)=a_0 + a_1 x + \cdots + a_n x^n, where a_0, \dots, a_n are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Lucas Theorem
In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometry, geometric relation between the root of a function, roots of a polynomial and the roots of its derivative . The set of roots of a real or complex polynomial is a set of point (geometry), points in the complex plane. The theorem states that the roots of all lie within the convex hull of the roots of , that is the smallest convex polygon containing the roots of . When has a single root then this convex hull is a single point and when the roots lie on a line (geometry), line then the convex hull is a line segment, segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle's theorem. Formal statement If is a (nonconstant) polynomial with complex coefficients, all root of a function, zeros of belong to the convex hull of the set of zeros of . Special cases It is easy to see that if P(x) = ax^2+bx+c is a second degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descartes' Rule Of Signs
In mathematics, Descartes' rule of signs, described by René Descartes in his ''La Géométrie'', counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of the polynomial's coefficients (omitting zero coefficients), and the difference between the root count and the sign change count is always even. In particular, when the number of sign changes is zero or one, then there are exactly zero or one positive roots. A linear fractional transformation of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of Budan's theorem and the Budan–Fourier theorem. Repeated division of an interval in two results in a set of disjoint intervals, each containing one root, and together listing all the roots. This approach is used in the fastest algorithms today for computer computation of real roots of polynomials (see real-root is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrally Closed Domain
In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' that is a root of a monic polynomial with coefficients in ''A,'' then ''x'' is itself an element of ''A.'' Many well-studied domains are integrally closed, as shown by the following chain of class inclusions: An explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well. A ring whose localizations at all prime ideals are integrally closed domains is a normal ring. Basic properties Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a field extension of ''K''. Then ''x''∈''L'' is integral over ''A'' if and only if it is algebraic over ''K'' and its minimal polynomial over ''K'' has coefficients in ''A''. In particular, this means that any elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Algebra
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one complex Zero of a function, root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field (mathematics), field of complex numbers is Algebraically closed field, algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, Degree of a polynomial, degree ''n'' polynomial with complex coefficients has, counted with Multiplicity (mathematics)#Multiplicity of a root of a polynomial, multiplicity, exactly ''n'' complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, it is not fundamental for modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest common divisor of and is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, . In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see ''Polynomial greatest common divisor'') and other commutative rings (see ' below). Overview Definition The ''greatest common divisor'' (GCD) of integers and , at least one of which is nonzero, is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
