AlgebraAlgebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Rational root theoremIn algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution x = p⁄q, written in lowest terms so that p and q are relatively prime, satisfies: p is an integer factor of the constant term a0, and q is an integer factor of the leading coefficient an.
Quadratic formulaIn elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Given a general quadratic equation of the form whose discriminant is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is: where the plus–minus symbol "±" indicates that the quadratic equation has two solutions.
FactorizationIn mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as whenever is not zero.
Algebraic equationIn mathematics, an algebraic equation or polynomial equation is an equation of the form where P is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term algebraic equation refers only to univariate equations, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables. In the case of several variables (the multivariate case), the term polynomial equation is usually preferred to algebraic equation.
Doubling the cubeDoubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other tools.
Quartic functionIn algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form where a ≠ 0. The derivative of a quartic function is a cubic function.
DiscriminantIn mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial is the quantity which appears under the square root in the quadratic formula.
Root of unityIn mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers.
Galois theoryIn mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials.
Abel–Ruffini theoremIn mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, (which was refined and completed in 1813 and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824.
Quintic functionIn mathematics, a quintic function is a function of the form where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function.
Angle trisectionAngle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle (that is, to construct an angle of 30 degrees).
Quadratic equationIn algebra, a quadratic equation () is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.
EquationIn mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true.
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Quartic equationIn mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is where a ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value). Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately.
Joseph-Louis LagrangeJoseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician, physicist and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
Casus irreducibilisIn algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843.
Cube rootIn mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other cube roots of 8 are and .
