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 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

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 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 or, equivalently, using Pascal's triangle. 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 triples: :For , let , , and ; then . * Binomials that are sums or differences of cubes 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