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 as:
Some examples of binomials are:
Operations on simple binomialsEdit
- The binomial x2 − y2 can be factored as the product of two other binomials:
- This is a special case of the more general formula:
- When working over the complex numbers, this can also be extended to:
- The product of a pair of linear binomials (ax + b) and (cx + d) is a trinomial:
- A binomial raised to the nth 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:
- 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 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: