1. Slide 11.5- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Binomial Theorem Learn to use Pascal’s Triangle to compute binomial coefficients. Learn to use Pascal’s Triangle to expand a binomial power. Learn to use the Binomial Theorem to expand a binomial power. Learn to find the coefficient of a term in a binomial expansion. SECTION 11.5 1 2 3 4

3. Slide 11.5- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PASCAL’S TRIANGLE When expanding (x + y) n the coefficients of each term can be determined using Pascal’s Triangle. The top of the triangle, that is, the first row, which contains only the number 1, represents the coefficients of (x + y) 0 and is referred to as Row 0. Row 2 represents the coefficients of (x + y) 1. Each row begins and ends with 1. Each entry of Pascal’s Triangle if found by adding the two neighboring entries in the previous row.

4. Slide 11.5- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PASCAL’S TRIANGLE An infinite sequence is a function whose

5. Slide 11.5- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Expand (4y – 2x) 5. Solution Row 5 of Pascal’s Triangle yields the binomial coefficients 1, 5, 10, 10, 5, 1. Replace x with 4y and y with –2x. EXAMPLE 1 Using Pascal’s Triangle to Expand a Binomial Power

6. Slide 11.5- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Using Pascal’s Triangle to Expand a Binomial Power Solution continued Expanding a difference results in alternating signs.

7. Slide 11.5- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF If r and n are integers with 0 ≤ r ≤ n, the we define

8. Slide 11.5- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE BINOMIAL THEOREM If n is a natural number, then the binomial expansion of (x + y) n is given by The coefficient of x n–r y r is

9. Slide 11.5- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PARTICULAR TERM IN A BINOMIAL EXPRESSION The term containing the factor x r in the expansion of (x + y) n is

10. Slide 11.5- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Finding a Particular Term in a Binomial Expansion Find the term containing x 10 in the expansion of (x + 2a) 15. Solution Use the formula for the term containing x r.

11. Slide 11.5- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Finding a Particular Term in a Binomial Expansion Solution continued