algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, a binomial is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

that is the

sum Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the additio ...

of two terms, each of which is a

monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...

. It is the simplest kind of sparse polynomial after the monomials.

Definition

A binomial is a polynomial which is the sum of two

s. A binomial in a single indeterminate (also known as a

univariate In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariat ...

binomial) can be written in the form :

a x^m - bx^n \,,

where and are

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...

s, and and are distinct

nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s and is a symbol which is called an

indeterminate Indeterminate may refer to: In mathematics * Indeterminate (variable), a symbol that is treated as a variable * Indeterminate system, a system of simultaneous equations that has more than one solution * Indeterminate equation, an equation that ha ...

or, for historical reasons, a variable. In the context of

Laurent polynomial In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' f ...

s, 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 can be

factored In 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 ...

as the product of two other binomials: ::

x^2 - y^2 = (x - y)(x + y).

:This is a

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...

of the more general formula: ::

x^ - y^ = (x - y)\sum_^ x^\,y^.

:When working over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, this can also be extended to: ::

x^2 + y^2 = x^2 - (iy)^2 = (x - iy)(x + iy).

*The product of a pair of linear binomials and is a trinomial: ::

(ax+b)(cx+d) = acx^2+(ad+bc)x+bd.

*A binomial raised to the ^th power, represented as can be expanded by means of the

binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...

or, equivalently, using

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, althoug ...

. For example, the square of the binomial is equal to the sum of the squares of the two terms and twice the product of the terms, that is: ::

(x + y)^2 = x^2 + 2xy + y^2.

:The numbers (1, 2, 1) appearing as multipliers for the terms in this expansion are

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 t ...

s two rows down from the top of Pascal's triangle. The expansion of the ^th power uses the numbers rows down from the top of the triangle. *An application of above formula for the square of a binomial is the "-formula" for generating

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

s: :For , let , , and ; then . * Binomials that are sums or differences of cubes can be factored into lower-order polynomials as follows: ::

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

Notes

References

* {{polynomials Algebra Factorial and binomial topics

Definition

Examples

Operations on simple binomials

See also

Notes

References