Binomial (polynomial): Difference between revisions

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| doi =
| accessdate = 29 March 2011}}</ref> It is the simplest kind of polynomial after the monomials.
 
==Definition==
A binomial is a polynomial which is the sum of two [[monomial]]s. A binomial in a single indeterminate (also known as a [[univariate]] binomial) can be written in the form
:<math>a x^m - bx^n \,,</math>
where {{math|''a''}} and {{math|''b''}} are [[number]]s, and {{math|''m''}} and {{math|''n''}} are distinct [[nonnegative integer]]s and {{math|''x''}} is a symbol which is called an [[indeterminate (variable)|indeterminate]] or, for historical reasons, a [[variable (mathematics)|variable]]. In the context of [[Laurent polynomial]]s, a ''Laurent binomial'', often simply called a ''binomial'', is similarly defined, but the exponents {{math|''m''}} and {{math|''n''}} may be negative.
 
More generally, a binomial may be written<ref name=Sturmfels62>{{Cite journal
| last = Sturmfels
| first = Bernd
| authorlink = Bernd Sturmfels
| journal = CBMS Regional Conference Series in Mathematics
| title = Solving Systems of Polynomial Equations
| publisher = Conference Board of the Mathematical Sciences
| issue = 97
| page = 62
| year = 2002
| url = https://books.google.com/books?id=N9c8bWxkz9gC
| accessdate = 21 March 2014}}</ref> as:
:<math>a x_1^{n_1}\dotsb x_i^{n_i} - b x_1^{m_1}\dotsb x_i^{m_i}</math>
 
Some examples of binomials are:
:<math>3x - 2x^2</math>
:<math>xy + yx^2</math>
:<math>0.9 x^3 + \pi y^2</math>
 
==Operations on simple binomials==