Difference between revisions of "Binomial Theorem"
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==Generalization== | ==Generalization== | ||
| − | The Binomial Theorem was generalized by [[Isaac Newton]], who used an [[infinite]] [[series]] to allow for complex [[exponent]]s | + | The Binomial Theorem was generalized by [[Isaac Newton]], who used an [[infinite]] [[series]] to allow for complex [[exponent]]s: For any [[real]] or [[complex]] <math>a</math>, <math>b</math>, and <math>r</math>, |
<center><math>(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k</math></center> | <center><math>(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k</math></center> | ||
Revision as of 18:57, 22 April 2008
The Binomial Theorem states that for real or complex 
, 
, and non-negative integer 
,
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 
.
Generalization
The Binomial Theorem was generalized by Isaac Newton, who used an infinite series to allow for complex exponents: For any real or complex , , and 
,
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.
