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

#### Guidance

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.

**Solution: **

**
**

**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:

#### Vocabulary

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

**is:**

*Binomial Theorem*

#### 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.

**Answers:**

1.

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 .

#### Practice

Expand each of the following binomials using the Binomial Theorem.

1.

2.

3.

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.

10.

11.

12.

Collapse the following polynomials using the Binomial Theorem.

13.

14.

15.