Difference between revisions of "Binomial Theorem"
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==The Theorem== | ==The Theorem== | ||
| − | First discovered by [[Newton]], the '''Binomial Theorem''' states that for [[real number | real]] or [[complex number |complex]] ''a'',''b'',<br><math>(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}</math>. | + | First discovered by [[Isaac Newton]], the '''Binomial Theorem''' states that for [[real number | real]] or [[complex number |complex]] ''a'',''b'',<br><math>(a+b)^n = \sum_{k=0}^{n}{n \choose k}\cdot a^k\cdot b^{n-k}</math>. |
This may be easily shown for the [[integer]]s:<br> | This may be easily shown for the [[integer]]s:<br> | ||
Revision as of 15:08, 18 July 2006
The Theorem
First discovered by Isaac Newton, the Binomial Theorem states that for real or complex a,b,
.
This may be easily shown for the integers:
.
Repeatedly using the distributive property, we see that for a term 
, we must choose 
of the 
terms to contribute an 
to the term, and then each of the other 
terms of the product must contribute a 
. Thus, the coefficient of is 
. Extending this to all possible values of from 
to , we see that .
Usage
Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial 
, one could factor it as such: 
. It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients.
