Binomial

# Binomial

Overview
In algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, a binomial is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

with two terms —the sum of two monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s—often bound by parenthesis or brackets when operated upon. It is the simplest kind of polynomial after the monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s.
Discussion

Encyclopedia
In algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, a binomial is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

with two terms —the sum of two monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s—often bound by parenthesis or brackets when operated upon. It is the simplest kind of polynomial after the monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s.

## Operations on simple binomials

• The binomial can be factored as the product of two other binomials:
This is a special case of the more general formula: .

• The product of a pair of linear binomials and is:

• A binomial raised to the nth power
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

, represented as
can be expanded by means of the binomial theorem
Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

or, equivalently, using Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

. Taking a simple example, the perfect square
Perfect square
- Mathematics :* Square number, a number that is a product of some integer with itself, e.g. 9 is a square number, its square root is 3* Perfect square dissection, a dissection of a geometric square into smaller squares, all of different sizes...

binomial can be found by squaring the first term, adding twice the product of the first and second terms and finally adding the square of the second term, to give .

• A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating Pythagorean triple
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, 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 k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...

s: for m < n, let , , , then .

• Binomial theorem
Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

• Completing the square
Completing the square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...

• Binomial distribution
• Binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

• Binomial-QMF
Binomial-QMF
Orthonormal binomial quadrature mirror filter bank with perfect reconstruction was designed by Ali Akansu, et al. published in 1990 using the family of binomial polynomials for subband decomposition of discrete-time signals....

(Daubechies Wavelet Filters)
• The list of factorial and binomial topics contains a large number of related links.
• Binomial series
Binomial series
In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f =  α, where is an arbitrary complex number...