In algebra, a **binomial** is a polynomial that is the sum of two terms, each of which is a monomial.^{[1]} It is the simplest kind of 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

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:

Some examples of binomials are:

## Operations on simple binomials

- The binomial
*x*^{2}−*y*^{2}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
*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:

- 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*=*n*^{2}−*m*^{2},*b*= 2*mn*, and*c*=*n*^{2}+*m*^{2}; then*a*^{2}+*b*^{2}=*c*^{2}.

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

## See also

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

## Notes

**^**Weisstein, Eric. "Binomial". Wolfram MathWorld. Retrieved 29 March 2011.**^**Sturmfels, Bernd (2002). "Solving Systems of Polynomial Equations".*CBMS Regional Conference Series in Mathematics*. Conference Board of the Mathematical Sciences (97): 62. Retrieved 21 March 2014.

## References

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