# Binomial Theorem

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.