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 ope ...

, a binomial is a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

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 a power product or primitive monomial, is a product of powers of variables with n ...

. 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 In mathematics, a univariate object is an expression (mathematics), expression, equation, function (mathematics), function or polynomial involving only one Variable (mathematics), variable. Objects involving more than one variable are ''wikt:multi ...

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 most basic 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 can ...

s, and and are distinct non-negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s 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)(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 .Brown, James Robert.� ...

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, the power expands into a polynomial with terms of the form , where the exponents and a ...

or, equivalently, using

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 Bla ...

. For example, the

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

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 the

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 the 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 , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...

s: :For , let , , and ; then . * Binomials that are sums or differences of

cubes A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

can be factored into smaller- degree 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 Polynomials Factorial and binomial topics

Definition

Examples

Operations on simple binomials

See also

Notes

References