Binomial Theorem

Pronunciation: /baɪˈnoʊ mi əl ˈθiər əm/ Explain

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. Merriam-Webster. Last Accessed 8/6/2018. 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. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 8/6/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n53/mode/1up/search/binomial. Buy the book

More Information

  • McAdams, David E.. Binomial. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/22/2018. http://www.allmathwords.org/en/b/binomial.html.

Cite this article as:

McAdams, David E. Binomial Theorem. 6/19/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/binomialtheorem.html.

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Revision History

6/22/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
12/3/2008: Changed equations to images. (McAdams, David E.)
6/28/2008: Replaced equations with dHot_Eqn.class (McAdams, David E.)
6/7/2008: Corrected link errors. Corrected spelling (McAdams, David E.)
4/18/2008: Initial version. (McAdams, David E.)

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