# Difference between revisions of "Binomial Theorem"

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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>, | 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> | ||

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+ | '''Proof''' | ||

+ | Consider the function <math>f(b)=(a+b)^r</math> for constants <math>a,r</math>. It is easy to see that <math>\frac{d^k}{db^k}f=r(r-1)\cdots(r-k+1)(a+b)^{r-k}</math>. Then, we have <math>\frac{d^k}{db^k}f(0)=r(r-1)\cdots(r-k+1)a^{r-k}</math>. So, the Taylor Series for <math>f(b)</math> is | ||

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+ | <math>(a+b)^k=\sum_{k=0}^\infty \frac{r(r-1)\cdots(r-k+1)a^{r-k}b^k}{k!}=\sum_{k=0}^\infty \binom{r}{k}a^{r-k}b^k</math>. | ||

==Usage== | ==Usage== |

## Revision as of 13:18, 3 August 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 ,

**Proof**
Consider the function for constants . It is easy to see that . Then, we have . So, the Taylor Series for is

.

## 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.