Binomial (polynomial)

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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.

Contents

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:

Examples

Operations on simple binomials

This is a special case of the more general formula:
When working over the complex numbers, this can also be extended to:
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.
For m < n, let a = n2m2, b = 2mn, and c = n2 + m2; then a2 + b2 = c2.

See also

Notes

  1. Weisstein, Eric. "Binomial". Wolfram MathWorld. Retrieved 29 March 2011.CS1 maint: discouraged parameter (link)
  2. Sturmfels, Bernd (2002). Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics. 97. American Mathematical Society. p. 62. ISBN   9780821889411.

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References