where and are numbers, and and are nonnegative integers and is a symbol which is called an indeterminate or, for historical reasons, a variable. In some contexts, the exponents and may be negative, in which case the monomial is a Laurent binomial.
More generally, a binomial may be written as:
Some examples of binomials are:
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: .
- This can also be extended to when working over the complex numbers
- The product of a pair of linear binomials and is a trinomial:
- A binomial raised to the nth power, represented as
- can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. 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 . 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 nth 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 , , , then .
- Binomials that are sums or differences of cubes can be factored into lower-order polynomials as follows:
- Completing the square
- Binomial distribution
- Binomial-QMF (Daubechies Wavelet Filters)
- The list of factorial and binomial topics contains a large number of related links.
- Binomial series