1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 11
Sequences, Induction, and Probability
11.5 The Binomial Theorem
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Evaluate a binomial coefficient.
Expand a binomial raised to a power.

3. Definition of a Binomial Coefficient

4. Example: Evaluating Binomial Coefficients
Evaluate each of the following:

5. Example: Evaluating Binomial Coefficients
Evaluate:

6. A Formula for Expanding Binomials: The Binomial Theorem

7. Example: Using the Binomial Theorem
Expand:

8. Example: Using the Binomial Theorem (continued)
Expand:

9. Introducing: Pascal’s Triangle
Take a moment to copy the first 6 rows. What patterns do you see?
Row 5
Row 6

10. The Binomial Theorem Use Pascal’s Triangle to expand (a + b)5.
Use the row that has 5 as its second number.
The exponents for a begin with 5 and decrease.
     1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5
The exponents for b begin with 0 and increase.
In its simplest form, the expansion is
a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5.
Row 5

11. The Binomial Theorem Use Pascal’s Triangle to expand (x – 3)4.
First write the pattern for raising a binomial to the fourth power.
   Coefficients from
Pascal’s Triangle.
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
Since (x – 3)4 = (x + (–3))4, substitute x for a and –3 for b.
(x + (–3))4 = x4 + 4x3(–3) + 6x2(–3)2 + 4x(–3)3 + (–3)4
= x4 – 12x3 + 54x2 – 108x + 81
The expansion of (x – 3)4 is x4 – 12x3 + 54x2 – 108x + 81.

12. The Binomial Theorem
For any positive integer, n

13. Let’s Try Some
Expand the following
a) (x-y5)3 b) (3x-2y)4

14. Let’s Try Some
Expand the following
(x-y5)3

15. Let’s Try Some
Expand the following
(3x-2y)4

16. Let’s Try Some
Expand the following
(3x-2y)4