# What is the binomial theorem?

##### 2 Answers

It is a method one may use to expand a binomial expression raised to a positive integer power as follows :

The combination notation used is defined as follows :

**Example :**

Expand

This is a binomial (2 terms) raised to an integer power, so the binomial theorem is valid and may be used as follows :

There is a simpler way of expanding a binomial that uses the binomial theorem but takes a more intuitive approach.

Instead of doing

In the case of

The row we want is

#1,5,10,10,5,1#

In order to deal with exponents, know that the exponent on the first term will start at

If there is a negative term they will alternate positive, negative, positive, negative, etc.

For

#1(2x)^5(3y)^0+5(2x)^4(3y)^1+10(2x)^3(3y)^2+10(2x)^2(3y)^3+5(2x)^1(3y)^4+1(2x)^0(3y)^5#

Note that anything to the

Simplified, this gives us

#32x^5+240x^4y+720x^3y^2+1080x^2y^3+810x^4+243y^5#