Binomial (polynomial)

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

${\displaystyle ax^{m}-bx^{n}\,,}$

where a and b are numbers, and m and n are distinct nonnegative integers and x 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 m and n may be negative.

More generally, a binomial may be written[2] as:

${\displaystyle ax_{1}^{n_{1}}\dotsb x_{i}^{n_{i}}-bx_{1}^{m_{1}}\dotsb x_{i}^{m_{i}}}$

Some examples of binomials are:

${\displaystyle 3x-2x^{2}}$

${\displaystyle xy+yx^{2}}$

${\displaystyle 0.9x^{3}+\pi y^{2}}$

• The binomial x² − y² can be factored as the product of two other binomials:

${\displaystyle x^{2}-y^{2}=(x-y)(x+y).}$

This is a special case of the more general formula:

${\displaystyle x^{n+1}-y^{n+1}=(x-y)\sum _{k=0}^{n}x^{k}\,y^{n-k}.}$

When working over the complex numbers, this can also be extended to:

${\displaystyle x^{2}+y^{2}=x^{2}-(iy)^{2}=(x-iy)(x+iy).}$

• The product of a pair of linear binomials (ax + b) and (cx + d) is a trinomial:

${\displaystyle (ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.}$

• A binomial raised to the n^th power, represented as (x + y)ⁿ can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square (x + y)² of the binomial (x + y) is equal to the sum of the squares of the two terms and twice the product of the terms, that is:

${\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}$

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 n^th power uses the numbers n rows down from the top of the triangle.

• An application of above formula for the square of a binomial is the "(m, n)-formula" for generating Pythagorean triples:

For m < n, let a = n² − m², b = 2mn, and c = n² + m²; then a² + b² = c².

• Binomials that are sums or differences of cubes can be factored into lower-order polynomials as follows:

${\displaystyle x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})}$

${\displaystyle x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})}$

• Completing the square
• Binomial distribution
• List of factorial and binomial topics (which contains a large number of related links)

• Bostock, L.; Chandler, S. (1978). Pure Mathematics 1. Oxford University Press. p. 36. ISBN 0-85950-092-6.