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''binomium of Newton''
 
''binomium of Newton''
  
 
The formula for the expansion of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the terms of the binomial:
 
The formula for the expansion of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the terms of the binomial:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
( z _ {1} + z _ {2} ) ^ {m\ } =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665002.png" /></td> </tr></table>
+
$$
 +
= \
 +
z _ {1}  ^ {m} + {
 +
\frac{m}{1!}
 +
} z _ {1} ^ {m - 1 }
 +
z _ {2} +
 +
\frac{m ( m - 1) }{2! }
 +
z _ {1} ^ {m -
 +
2 } z _ {2}  ^ {2} + \dots + z _ {2}  ^ {m\ } =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665003.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {k = 0 } ^ { m }  \left ( \begin{array}{c}
 +
m \\
 +
k
 +
\end{array}
 +
\right
 +
) z _ {1} ^ {m - k } z _ {2}  ^ {k} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665004.png" /></td> </tr></table>
+
$$
 +
\left (
 +
\begin{array}{c}
 +
m \\
 +
k
 +
\end{array}
 +
 
 +
\right )  = \
  
are the [[Binomial coefficients|binomial coefficients]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665005.png" /> terms formula (*) takes the form
+
\frac{m! }{k! ( m - k)! }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665006.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665007.png" /></td> </tr></table>
+
are the [[Binomial coefficients|binomial coefficients]]. For  $  n $
 +
terms formula (*) takes the form
  
For an arbitrary exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665008.png" />, real or even complex, the right-hand side of (*) is, generally speaking, a [[Binomial series|binomial series]].
+
$$
 +
( z _ {1} + \dots + z _ {n} ) ^ {m\ } =
 +
$$
  
The gradual mastering of binomial formulas, beginning with the simplest special cases (formulas for the  "square"  and the "cube of a sum" ) can be traced back to the 11th century. I. Newton's contribution, strictly speaking, lies in the discovery of the binomial series.
+
$$
 +
= \
 +
\sum _ {k _ {1} + \dots + k _ {n} = m }
 +
\frac{m! }{k _ {1} ! \dots
 +
k _ {n} ! }
 +
  z _ {1} ^ {k _ {1} } \dots z _ {n} ^ {k _ {n} } .
 +
$$
  
 +
For an arbitrary exponent  $  m $,
 +
real or even complex, the right-hand side of (*) is, generally speaking, a [[Binomial series|binomial series]].
  
 +
The gradual mastering of binomial formulas, beginning with the simplest special cases (formulas for the  "square"  and the  "cube of a sum" ) can be traced back to the 11th century. I. Newton's contribution, strictly speaking, lies in the discovery of the binomial series.
  
 
====Comments====
 
====Comments====
 
The coefficients
 
The coefficients
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066500/n0665009.png" /></td> </tr></table>
+
$$
 +
\left ( \begin{array}{c}
 +
m \\
 +
k _ {1} \dots k _ {n}
 +
\end{array}
 +
\right )  = \
 +
 
 +
\frac{m! }{k _ {1} ! \dots k _ {n} ! }
 +
,\ \
 +
k _ {1} + \dots + k _ {n} = m,
 +
$$
  
 
are called multinomial coefficients.
 
are called multinomial coefficients.

Latest revision as of 08:02, 6 June 2020


binomium of Newton

The formula for the expansion of an arbitrary positive integral power of a binomial in a polynomial arranged in powers of one of the terms of the binomial:

$$ \tag{* } ( z _ {1} + z _ {2} ) ^ {m\ } = $$

$$ = \ z _ {1} ^ {m} + { \frac{m}{1!} } z _ {1} ^ {m - 1 } z _ {2} + \frac{m ( m - 1) }{2! } z _ {1} ^ {m - 2 } z _ {2} ^ {2} + \dots + z _ {2} ^ {m\ } = $$

$$ = \ \sum _ {k = 0 } ^ { m } \left ( \begin{array}{c} m \\ k \end{array} \right ) z _ {1} ^ {m - k } z _ {2} ^ {k} , $$

where

$$ \left ( \begin{array}{c} m \\ k \end{array} \right ) = \ \frac{m! }{k! ( m - k)! } $$

are the binomial coefficients. For $ n $ terms formula (*) takes the form

$$ ( z _ {1} + \dots + z _ {n} ) ^ {m\ } = $$

$$ = \ \sum _ {k _ {1} + \dots + k _ {n} = m } \frac{m! }{k _ {1} ! \dots k _ {n} ! } z _ {1} ^ {k _ {1} } \dots z _ {n} ^ {k _ {n} } . $$

For an arbitrary exponent $ m $, real or even complex, the right-hand side of (*) is, generally speaking, a binomial series.

The gradual mastering of binomial formulas, beginning with the simplest special cases (formulas for the "square" and the "cube of a sum" ) can be traced back to the 11th century. I. Newton's contribution, strictly speaking, lies in the discovery of the binomial series.

Comments

The coefficients

$$ \left ( \begin{array}{c} m \\ k _ {1} \dots k _ {n} \end{array} \right ) = \ \frac{m! }{k _ {1} ! \dots k _ {n} ! } ,\ \ k _ {1} + \dots + k _ {n} = m, $$

are called multinomial coefficients.

How to Cite This Entry:
Newton binomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton_binomial&oldid=13002
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article