algebra Algebra () is one of the 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 mathematics. Elementary ...

, 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 example ...

that is the sum 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 expone ...

. It is the simplest kind of

sparse polynomial In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. Examples include *monomials, polynomials with only one term, * binomials, po ...

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 multivariate ...

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 c ...

s, and and are distinct nonnegative integers and is a symbol which is called an indeterminate or, for historical reasons, a

variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...

. 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 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 i ...

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 fo ...

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 In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. Examples of trinomial expressions # 3x + 5y + 8z with x, y, z variables # 3t + 9s^2 + 3y^3 with t, s, y variables # 3ts + 9t + 5s with t, s variables # ...

: ::

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

*A binomial raised to the ^th

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

, 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, although o ...

. 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 coefficients 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