1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

2. Chapter 12 Sequences, Series, and the Binomial Theorem

3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 12.5 The Binomial Theorem

4. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall (a + b) 0 = 1 (a + b) 1 = a + b (a + b) 2 = a 2 + 2ab + b 2 (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4a 1 b 3 + b 4 (a + b) 5 = a 5 + 5a 4 b + 10a 3 b 2 + 10a 2 b 3 + 5ab 4 + b 5 Expanding a binomial such as (a + b) n means to write the factored form as a sum. 1 term 2 terms 3 terms 4 terms 5 terms 6 terms

5. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall (a + b) 0 = 1 (a + b) 1 = a + b (a + b) 2 = a 2 + 2ab + b 2 (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4a 1 b 3 + b 4 (a + b) 5 = a 5 + 5a 4 b + 10a 3 b 2 + 10a 2 b 3 + 5ab 4 + b 5 1.The expansion of (a + b) n contains n + 1 terms. 2.The first term is a n and the last term is b n. 3.The powers of a decrease by 1 for each term; the powers of b increase by 1 for each term. 4.The sum of the exponents of a and b is n. Notice the following patterns

6. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall There are also patterns in the coefficients of the terms. When written in a triangular array, the coefficients are called Pascal’s triangle.

7. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 (a + b) 0 (a + b) 1 (a + b) 2 (a + b) 3 (a + b) 4 (a + b) 5 1612015 6 Add the consecutive numbers in the row for n = 5 and write each sum “between and below” the pair. 1 5 6 Pascal’s Triangle

8. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Expand (a + b) 7. Use n = 7 row of Pascal’s triangle as the coefficients and the noted patterns. (a + b) 7 = a 7 + 7a 6 b + 21a 5 b 2 + 35a 4 b 3 + 35a 3 b 4 n = 71 7 21 35 35 21 7 1 + 21a 2 b 5 + 7ab 6 + b 7 Example Solution

9. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall An alternative method for determining the coefficients of (a + b) n is based on using factorials. The factorial of n, written n! (read “n factorial”), is the product of the first n consecutive natural numbers. Factorial of n: n! If n is a natural number, then n! = n(n – 1)(n – 2)(n – 3)... ∙ 3 ∙ 2 ∙ 1. The factorial of 0, written 0!, is defined to be 1.

10. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Evaluate each expression. Example Solution

11. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall It can be proved that the coefficients of terms in the expansion of (a + b) n can be expressed in terms of factorials. Following the earlier patterns and using the factorial expressions of the coefficients, we have the binomial theorem. Binomial Theorem If n is a positive integer, then

12. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the binomial theorem to expand (x + 3) 4. Example Solution

13. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the binomial theorem to expand (3a – 5b) 6. Example Solution

14. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall (r + 1) st Term in a Binomial Expansion The (r + 1) st term of the expansion of (a + b) n is

15. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the ninth term in the expansion of (3x – 5y) 10. n = 10, a = 3x, b = – 5y, r + 1 = 9, therefore r = 8 Example Solution