Binomial (polynomial): Difference between revisions

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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</math>''}} and <{{math>|''n</math>''}} may be negative.
 
More generally, a binomial may be written<ref name=Sturmfels62>{{Cite journal
 
==Operations on simple binomials==
*The binomial <{{math> |''x^''<sup>2</sup> - ''y^''<sup>2 </mathsup>}} can be factored as the product of two other binomials.:
::<math> x^2 - y^2 = (x + y)(x - y). </math>
:This is a special case of the more general formula: <math> x^{n+1} - y^{n+1} = (x - y)\sum_{k=0}^{n} x^{k}\,y^{n-k}</math>.
:This can also be extended to :<math> x^2 {n+1} - y^2{n+1} = (x^2 - (iyy)\sum_{k=0}^2{n} = (x ^{k}\,y^{n- iy)(x + iy) k}.</math> when working over the complex numbers
:When working over the complex numbers, this can also be extended to
::<math> x^2 + y^2 = x^2 - (iy)^2 = (x - iy)(x + iy). </math>
 
*The product of a pair of linear binomials <math>(ax+b)</math> and <math>(cx+d)</math> is a [[trinomial]]: