Binomial Theorem

Pronunciation: /baɪˈnoʊ mi əl ˈθiər əm/ ?

The binomial theorem states that when raising a binomial to an integer power, the binomial coefficients can be calculated using the combination

choose(n,m) = (n!)/((n-k)!k!)
where n is the integer power, and k is the order of the coefficient.[1] This is expressed mathematically with the expression
n=sum from k=0 to n (choose(n,k)x^(n-k)a^(k))
Start with the simple binomial
(x+y)^2
In this expression
n=2
So the first term is
choose(2,0)x^(2-0)y^0=x^2
The next two terms are
choose(2,1)x^(2-1)y^1=2xy
and
choose(2,2)x^(2-2)y^2=y^2
This gives a result of
(x+y)^2=x^2+2xy+y^2
You can use long multiplication to verify this result.

References

  1. binomial theorem. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2009-03-12). http://www.merriam-webster.com/dictionary/binomial theorem.
  2. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically, 2nd edition, pp 44-53. D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 2009-12-19). http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n53/mode/1up/search/binomial.

More Information

  • McAdams, David. Binomial. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Binomial.

Cite this article as:


Binomial Theorem. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/binomialtheorem.html.

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2009-12-19: Added "References" (McAdams, David.)
2008-12-03: Changed equations to images (McAdams, David.)
2008-06-28: Replaced equations with dHot_Eqn.class (McAdams, David.)
2008-06-07: Corrected link errors. Corrected spelling (McAdams, David.)
2008-04-18: Initial version (McAdams, David.)

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