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Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials. Definition A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form :a x^m - bx^n , where and are numbers, and and are distinct non-negative integers and is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a ''Laurent binomial'', often simply called a ''binomial'', is similarly defined, but the exponents and may be negative. More generally, a binomial may be written as: :a\, x_1^\dotsb x_i^ - b\, x_1^\dotsb x_i^ Examples :3x - 2x^2 :xy + yx^2 :0.9 x^3 + \pi y^2 :2 x^3 + 7 Operations on simple binomials *The binomial , the difference of two squares, can be factored as the product of two other binomials: :: x^2 - y^2 = (x - y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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List Of Factorial And Binomial Topics
{{Short description, none This is a list of factorial and binomial topics in mathematics. See also binomial (other). * Abel's binomial theorem *Alternating factorial *Antichain *Beta function * Bhargava factorial *Binomial coefficient **Pascal's triangle *Binomial distribution *Binomial proportion confidence interval * Binomial-QMF ( Daubechies wavelet filters) *Binomial series *Binomial theorem *Binomial transform *Binomial type * Carlson's theorem *Catalan number ** Fuss–Catalan number *Central binomial coefficient *Combination *Combinatorial number system *De Polignac's formula *Difference operator * Difference polynomials *Digamma function * Egorychev method * Erdős–Ko–Rado theorem *Euler–Mascheroni constant *Faà di Bruno's formula *Factorial * Factorial moment *Factorial number system *Factorial prime * Factoriangular number *Gamma distribution *Gamma function *Gaussian binomial coefficient * Gould's sequence * Hyperfactorial *Hypergeometric distribution * Hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory), experiments, each asking a yes–no question, and each with its own Boolean-valued function, Boolean-valued outcome (probability), outcome: ''success'' (with probability ) or ''failure'' (with probability ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., , the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size drawn with replacement from a population of size . If the sampling is carried out without replacement, the draws ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Completing The Square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By subsequently isolating and taking the square root, a quadratic problem can be reduced to a linear problem. The name ''completing the square'' comes from a geometrical picture in which represents an unknown length. Then the quantity represents the area of a square of side and the quantity represents the area of a pair of Congruence (geometry), congruent rectangles with sides and . To this square and pair of rectangles one more square is added, of side length . This crucial step ''completes'' a larger square of side length . Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian Empire, Old Babylonian clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see Order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cube (algebra)
In arithmetic and algebra, the cube of a number is its third exponentiation, power, that is, the result of multiplying three instances of together. The cube of a number is denoted , using a superscript 3, for example . The cube Mathematical operation, operation can also be defined for any other expression (mathematics), mathematical expression, for example . The cube is also the number multiplied by its square (algebra), square: :. The ''cube function'' is the function (mathematics), function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a Cube (geometry), geometric cube is the cube of its side length, giving rise to the name. The Inverse function, inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of a function, graph of the cube function is known as the cubic para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pythagorean Triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements). The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 ( caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is '' quadratic''. The square of an integer may also be called a '' square number'' or a ''perfect square''. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pascal's Triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |