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Binomial Theorem and Expansions

Expansion of binomials raised to a power using combinations.

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Binomial Theorem

The Binomial Theorem tells you how to expand a binomial such as  without having to compute the repeated distribution. What is the expanded version of ?

Watch This

http://www.youtube.com/watch?v=YxysKtqpbVI James Sousa: The Binomial Theorem Using Combinations

http://www.youtube.com/watch?v=NLQmQGA4a3M James Sousa: The Binominal Theorem Using Pascal’s Triangle


The Binomial Theorem states:

Writing out a few terms of the summation symbol helps you to understand how this theorem works:

Going from one term to the next in the expansion, you should notice that the exponents of  decrease while the exponents of  increase. You should also notice that the coefficients of each term are combinations. Recall that  is the number of ways to choose  objects from a set of  objects. 

Another way to think about the coefficients in the Binomial Theorem is that they are the numbers from Pascal’s Triangle. Look at the expansions of  below and notice how the coefficients of the terms are the numbers in Pascal’s Triangle.

Be extremely careful when working with binomials of the form . You need to remember to capture the negative with the second term as you write out the expansion: .

Example A

Expand the following binomial using the Binomial Theorem. 


Example B

What is the coefficient of the term  in the expansion of the binomial ?

Solution: The Binomial Theorem allows you to calculate just the coefficient you need.

Example C

What is the coefficient of  in the expansion of ?

Solution: For this problem you should calculate the whole term, since the 3 and the 4 in will impact the coefficient of as well. . The coefficient is 20,412.

Concept Problem Revisited

The expanded version of   is:


The Binomial Theorem is a theorem that states how to expand binomials that are raised to a power using combinations. The Binomial Theorem is:

Guided Practice

1. What is the coefficient of  in the expansion of ?

2. Compute the following summation.

3. Collapse the following polynomial using the Binomial Theorem.



2. This is asking for , which are the sum of all the coefficients of .

3. Since the last term is -1 and the power on the first term is a 5 you can conclude that the second half of the binomial is . The first term is positive and , so the first term in the binomial must be . The binomial is .


Expand each of the following binomials using the Binomial Theorem.




4. What is the coefficient of  in ?

5. What is the coefficient of in ?

6. What is the coefficient of  in ?

7. What is the coefficient of in ?

8. What is the coefficient of  in ?

9. What is the coefficient of  in ?

Compute the following summations.




Collapse the following polynomials using the Binomial Theorem.







Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.

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