A General Binomial Theorem

How to deal with negative and fractional exponents

The Binomial Theorem is commonly stated in a way that works well for positive integer exponents.

How can we apply it when we have a fractional or negative exponent? For example:

The problem is with the coefficient, which we usually define using factorials.

When n is other than a non-negative integer, n! and (n-k)! present a problem. While we do have the Gamma Function to handle such animals, we can find a much simpler solution. All we need is elementary algebra.

Let’s look at a specific example, and then generalize.

From this one example, we can make the general observation.

Our new Binomial Theorem looks like this.

Let’s take this baby out for a spin.

n=½