# Difference between revisions of "Binomial Theorem"

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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> centered at <math>0</math> is | 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> centered at <math>0</math> is | ||

− | <cmath>(a+b)^ | + | <cmath>(a+b)^r=\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.</cmath> |

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

## Revision as of 03:43, 5 April 2017

The **Binomial Theorem** states that for real or complex , , and non-negative integer ,

where is a binomial coefficient. In other words, the coefficients when is expanded and like terms are collected are the same as the entries in the th row of Pascal's Triangle.

For example, , with coefficients , , , etc.

## Contents

## Proofs

There are a number of different ways to prove the Binomial Theorem, for example by a straightforward application of mathematical induction. The Binomial Theorem also has a nice combinatorial proof:

We can write . 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 the number of ways to choose objects from a set of size , or . Extending this to all possible values of from to , we see that , as claimed.

## Generalizations

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 centered at 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.