Editing Binomial (polynomial)

{{Short description|In mathematics, a polynomial with two terms}}
In [[algebra]], a '''binomial''' is a [[polynomial]] that is the sum of two terms, each of which is a [[monomial]].<ref>{{mathworld|id=Binomial|title=Binomial}}</ref> It is the simplest kind of a [[sparse polynomial]] after the monomials.

==Definition==
A binomial is a polynomial which is the sum of two monomials. A binomial in a single [[indeterminate (variable)|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 non-negative [[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 book
  | last = Sturmfels
  | first = Bernd
  | author-link = Bernd Sturmfels
  | series = CBMS Regional Conference Series in Mathematics
  | title = Solving Systems of Polynomial Equations
  | volume = 97
  | page = 62
  | year = 2002 |isbn=9780821889411 |publisher=American Mathematical Society
  | url = https://books.google.com/books?id=N9c8bWxkz9gC
  }}</ref> as:
:<math>a\, x_1^{n_1}\dotsb x_i^{n_i} - b\, x_1^{m_1}\dotsb x_i^{m_i}</math>

==Examples==
:<math>3x - 2x^2</math>
:<math>xy + yx^2</math>
:<math>0.9 x^3 + \pi y^2</math>
:<math>2 x^3 + 7</math>

==Operations on simple binomials==
*The binomial {{math|''x''<sup>2</sup> − ''y''<sup>2</sup>}}, the [[difference of two squares]], 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>
:When working over the [[complex number]]s, 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'')}} and {{math|(''cx'' + ''d''&thinsp;)}} is a [[trinomial]]:
::<math> (ax+b)(cx+d) = acx^2+(ad+bc)x+bd.</math>
*A binomial raised to the {{math|''n''}}<sup>th</sup> [[Exponentiation|power]], represented as {{math|(''x'' + ''y'')<sup>''n''</sup>}} can be expanded by means of the [[binomial theorem]] or, equivalently, using [[Pascal's triangle]]. For example, the [[square (algebra)|square]] {{math|(''x'' + ''y'')<sup>2</sup>}} of the binomial {{math|(''x'' + ''y'')}} is equal to the sum of the squares of the two terms and twice the product of the terms, that is:
::<math> (x + y)^2 = x^2 + 2xy + y^2.</math>
:The numbers (1,&nbsp;2,&nbsp;1) appearing as multipliers for the terms in this expansion are the [[binomial coefficient]]s two rows down from the top of Pascal's triangle. The expansion of the {{math|''n''}}<sup>th</sup> power uses the numbers {{math|''n''}} rows down from the top of the triangle.
*An application of the above formula for the square of a binomial is the "{{math|(''m'',&thinsp;''n'')}}-formula" for generating [[Pythagorean triple]]s:
:For {{math|''m'' < ''n''}}, let {{math|''a'' {{=}} ''n''<sup>2</sup> − ''m''<sup>2</sup>}}, {{math|''b'' {{=}} 2''mn''}}, and {{math|''c'' {{=}} ''n''<sup>2</sup> + ''m''<sup>2</sup>}}; then {{math|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>}}.
* Binomials that are sums or differences of [[cube (algebra)|cubes]] can be factored into smaller-[[degree of a polynomial|degree]] polynomials as follows:
::<math> x^3 + y^3 = (x + y)(x^2 - xy + y^2) </math>
::<math> x^3 - y^3 = (x - y)(x^2 + xy + y^2) </math>

==See also==
*[[Completing the square]]
*[[Binomial distribution]]
*[[List of factorial and binomial topics]] (which contains a large number of related links)

== Notes ==
{{reflist}}

==References==
* {{cite book |first1=L. |last1=Bostock |author-link1=Linda Bostock |first2=S. |last2=Chandler |author-link2=Sue Chandler |title=Pure Mathematics 1 |isbn=0-85950-092-6 |publisher=[[Oxford University Press]] |date=1978 |page=36}}

{{polynomials}}

[[Category:Polynomials]]
[[Category:Factorial and binomial topics]]