# Difference between revisions of "Binomial Theorem"

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

Many [[factoring | factorizations]] involve complicated [[polynomial]]s with [[binomial coefficient]]s. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>, one could factor it as such: <math> x(x^4+4x^3+6x^2+4x+1)=x(x+1)^{4}</math>. It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients. | Many [[factoring | factorizations]] involve complicated [[polynomial]]s with [[binomial coefficient]]s. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>, one could factor it as such: <math> x(x^4+4x^3+6x^2+4x+1)=x(x+1)^{4}</math>. It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients. | ||

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+ | In addition, the expansion of a polynomial such as <math>(x+y)^n</math> will have coefficients corresponding to the <math>nth</math> row of [[Pascal's Triangle]]. For example, <math>(x+1)^5</math> = <math>x^5+5x^4+10x^3+10x^2+5x+1</math>, and the integers <math>1</math>, <math>5</math>, <math>10</math>, <math>10</math>, <math>5</math>, and <math>1</math> make up the 5th row of Pascal's Triangle. | ||

==See also== | ==See also== |

## Revision as of 01:07, 11 December 2009

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

where is a binomial coefficient. This result has a nice combinatorial proof: . 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 .

## Contents

## 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.

In addition, the expansion of a polynomial such as will have coefficients corresponding to the row of Pascal's Triangle. For example, = , and the integers , , , , , and make up the 5th row of Pascal's Triangle.