Binomial (polynomial)

<h2 id="definition">Definition</h2>
A binomial is a polynomial which is the sum of two monomials. A binomial in a single <a href="/facts/Indeterminate_(variable)/R7QAHS4j">indeterminate</a> (also known as a <a href="/facts/Univariate/PYwC7u5H">univariate</a> binomial) can be written in the form

a
        
          x
          
            m
          
        
        −
        b
        
          x
          
            n
          
        
        ,
      
    
    {\displaystyle ax^{m}-bx^{n},}

where a and b are <a href="/facts/Number/auQSAE2m">numbers</a>, and m and n are distinct non-negative <a href="/facts/Integer/PUwwrawx">integers</a> and x is a symbol which is called an <a href="/facts/Indeterminate_(variable)/R7QAHS4j">indeterminate</a> or, for historical reasons, a <a href="/facts/Variable_(mathematics)/VbQlrvKz">variable</a>. In the context of <a href="/facts/Laurent_polynomial/ESxjzdmw">Laurent polynomials</a>, 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<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> as:

a
        
        
          x
          
            1
          
          
            
              n
              
                1
              
            
          
        
        ⋯
        
          x
          
            i
          
          
            
              n
              
                i
              
            
          
        
        −
        b
        
        
          x
          
            1
          
          
            
              m
              
                1
              
            
          
        
        ⋯
        
          x
          
            i
          
          
            
              m
              
                i
              
            
          
        
      
    
    {\displaystyle a\,x_{1}^{n_{1}}\dotsb x_{i}^{n_{i}}-b\,x_{1}^{m_{1}}\dotsb x_{i}^{m_{i}}}

<h2 id="examples">Examples</h2>

3
        x
        −
        2
        
          x
          
            2
          
        
      
    
    {\displaystyle 3x-2x^{2}}

x
        y
        +
        y
        
          x
          
            2
          
        
      
    
    {\displaystyle xy+yx^{2}}

0.9
        
          x
          
            3
          
        
        +
        π
        
          y
          
            2
          
        
      
    
    {\displaystyle 0.9x^{3}+\pi y^{2}}

2
        
          x
          
            3
          
        
        +
        7
      
    
    {\displaystyle 2x^{3}+7}

<h2 id="operations-on-simple-binomials">Operations on simple binomials</h2>
<ul><li>The binomial x2 − y2, the <a href="/facts/Difference_of_two_squares/3gkVr71S">difference of two squares</a>, can be <a href="/facts/Factored/tXrBXFh1">factored</a> as the product of two other binomials:</li></ul>

x
          
            2
          
        
        −
        
          y
          
            2
          
        
        =
        (
        x
        −
        y
        )
        (
        x
        +
        y
        )
        .
      
    
    {\displaystyle x^{2}-y^{2}=(x-y)(x+y).}

This is a <a href="/facts/Special_case/bKaJ85XJ">special case</a> of the more general formula:

x
          
            n
            +
            1
          
        
        −
        
          y
          
            n
            +
            1
          
        
        =
        (
        x
        −
        y
        )
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        
          x
          
            k
          
        
        
          y
          
            n
            −
            k
          
        
        .
      
    
    {\displaystyle x^{n+1}-y^{n+1}=(x-y)\sum _{k=0}^{n}x^{k}y^{n-k}.}

When working over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, this can also be extended to:

x
          
            2
          
        
        +
        
          y
          
            2
          
        
        =
        
          x
          
            2
          
        
        −
        (
        i
        y
        
          )
          
            2
          
        
        =
        (
        x
        −
        i
        y
        )
        (
        x
        +
        i
        y
        )
        .
      
    
    {\displaystyle x^{2}+y^{2}=x^{2}-(iy)^{2}=(x-iy)(x+iy).}

<ul><li>The product of a pair of linear binomials (ax + b) and (cx + d ) is a <a href="/facts/Trinomial/nVXhBpbH">trinomial</a>:</li></ul>

(
        a
        x
        +
        b
        )
        (
        c
        x
        +
        d
        )
        =
        a
        c
        
          x
          
            2
          
        
        +
        (
        a
        d
        +
        b
        c
        )
        x
        +
        b
        d
        .
      
    
    {\displaystyle (ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.}

<ul><li>A binomial raised to the nth <a href="/facts/Exponentiation/pac6QY2g">power</a>, represented as (x + y)n can be expanded by means of the <a href="/facts/Binomial_theorem/hd4SmkUd">binomial theorem</a> or, equivalently, using <a href="/facts/Pascal%2527s_triangle/tb6BKXbc">Pascal's triangle</a>. For example, the <a href="/facts/Square_(algebra)/Jzpq0R0z">square</a> (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:</li></ul>

(
        x
        +
        y
        
          )
          
            2
          
        
        =
        
          x
          
            2
          
        
        +
        2
        x
        y
        +
        
          y
          
            2
          
        
        .
      
    
    {\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}

The numbers (1, 2, 1) appearing as multipliers for the terms in this expansion are the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficients</a> 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.
<ul><li>An application of the above formula for the square of a binomial is the "(m, n)-formula" for generating <a href="/facts/Pythagorean_triple/I3IJgWhf">Pythagorean triples</a>:</li></ul>
For m < n, let a = n2 − m2, b = 2mn, and c = n2 + m2; then a2 + b2 = c2.
<ul><li>Binomials that are sums or differences of <a href="/facts/Cube_(algebra)/SqmlEEog">cubes</a> can be factored into smaller-<a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> polynomials as follows:</li></ul>

x
          
            3
          
        
        +
        
          y
          
            3
          
        
        =
        (
        x
        +
        y
        )
        (
        
          x
          
            2
          
        
        −
        x
        y
        +
        
          y
          
            2
          
        
        )
      
    
    {\displaystyle x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})}

x
          
            3
          
        
        −
        
          y
          
            3
          
        
        =
        (
        x
        −
        y
        )
        (
        
          x
          
            2
          
        
        +
        x
        y
        +
        
          y
          
            2
          
        
        )
      
    
    {\displaystyle x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Completing_the_square/SN6xA8sc">Completing the square</a></li>
<li><a href="/facts/Binomial_distribution/UMoFMjDj">Binomial distribution</a></li>
<li><a href="/facts/List_of_factorial_and_binomial_topics/e3rKQq7R">List of factorial and binomial topics</a> (which contains a large number of related links)</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Linda_Bostock/vf0K5J78">Bostock, L.</a>; <a href="/facts/Sue_Chandler/vf0K5J78">Chandler, S.</a> (1978). Pure Mathematics 1. <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>. p. 36. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-85950-092-6.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weisstein, Eric W. "Binomial". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Sturmfels, Bernd (2002). Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics. Vol. 97. American Mathematical Society. p. 62. ISBN 9780821889411. <a href="9780821889411" target="_blank">9780821889411</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Binomial (polynomial) open-in-new

Binomial (polynomial)