Binomial Expansion

A binomial expression is the sum or difference of two terms, like x-y, 2a+3b. Expanding (x+1) by the power of 2 is simple, but when it comes to expanding to the power of larger numbers, you need to use other methods. One of the methods is using Pascal's Triangle.

Pascal's Triangle

Pascal's triangle is a special diagram created by Blaise Pascal, where two numbers on a row add up to the number below it.

What's special here is that the rows of numbers are the binomial coefficients.

To expand the binomial expression below, we have to look at the row that begins with 1 and 3, which is 1, 3, 3, 1. As the power of a decreases, the power of b increases.

The power of a goes from 3 to 0, and the power of b goes from 0 to 3.                      

The Binomial Theorem

The theorem is used when the power is large and where Pascal's triangle cannot be used. However, it can enable us to expand to powers of a positive whole number.

2! means 2 factorial, or 2 x 1. 5! = 5 x 4 x 3 x 2 x 1

We will use the binomial theorem to expand this binomial expression.

a will be substituted as 1, b will be x and n will be 2. Putting these values into the binomial theorem shows us that the binomial can be expanded this way.

Pictures from: http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-pascal-2009-1.pdf