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Binomial Theorem: Applications & Examples

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn how useful the Binomial Theorem is in helping you to easily find the answer to expanding a binomial expression to any power.

The Binomial Theorem

Let's start off by introducing the Binomial Theorem. This theorem is a very useful theorem and it helps you find the expansion of binomials raised to any power. It can help you find answers to binomial problems such as:

  • (3x + 4y)5
  • (10x - 2y)10
  • (x + y)12

Notice how these binomials aren't simply squared or tripled. No, these have much higher powers. Now, you could do these by hand, but the computations can get rather messy and hard to keep track of on paper. Just think of how many terms you get when you square a binomial. You'll have many more when you use powers of 4 or more. It can get messy. So, mathematicians came up with and proved the Binomial Theorem to solve these problems. Here's the theorem expressed in a formal way.


binomial theorem


The bit in parentheses is actually part of statistics and probability and it means n choose k. It uses factorials to figure out the number.


binomial theorem


Now, let's look at how you can use the Binomial Theorem with three different applications.

Positive Integral Index

Let's start with applying the Binomial Theorem to find the positive integral index. The positive integral index uses only positive powers, so all your n are positive integers. This is a straightforward application of the Binomial Theorem. You'll be following the formula just as you see it.

This is the Binomial Theorem used to expand this problem: (2x + y)4


binomial theorem


Everything was plugged into the problem and then evaluated. Pretty straightforward. Not as messy as expanding it by multiplying it out one term at a time and then combining all the like terms.

Rational Index

Another application of the Binomial Theorem is for the rational index. This is when you change the form of your binomial to a form like this:

  • (1 + x)n, where the absolute value of x is less than 1 and n can be either an integer or a fraction

You can get to this form by dividing your binomial by the a like this.

  • (a + b)5 => (1 + b / a)5

The absolute value of your x (in this case b / a) has to be less than 1 for this expansion formula to work.

Here is the formula to use for the rational index:


binomial theorem


To use this form of the Binomial Theorem, you have to make sure that the absolute value of x is less than 1. This is usually stated in the problem somewhere. This formula can be used for any power, integer, and fraction. As you can see, this gives you an infinite series.

Just like the Binomial Theorem for the positive integral index, you plug in your values and evaluate. The only additional thing that you may have to do is to figure out what your infinite series converges to (this won't be discussed in this lesson though).

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