1. The Binomial Theorem Section 9.2a!!!

2. Powers of Binomials Let’s expand (a + b) for n = 0, 1, 2, 3, 4, and 5: n Do you see a pattern with the binomial coefficients ???

3. Definition: Binomial Coefficient The binomial coefficients that appear in the expansion of (a + b) are the values of C for r = 0, 1, 2,…,n. n nr Recall that a classical notation for C (especially in nr the context of binomial coefficients) is. n r Both notations are read “n choose r.”

4. Finding Binomial Coefficients Expand (a + b), using a calculator to compute the binomial coefficients. 5 Enter 5 C {0, 1, 2, 3, 4, 5} into your calculator… nr The calculator returns the list {1, 5, 10, 10, 5, 1}…

5. Pascal’s Triangle To obtain this famous figure, take only the positive coefficients from the “triangular array” from our first example of the day: 1 11 211 3311 64411 10105511 Row 0 Row 1

6. Extending Pascal’s Triangle Show how row 5 of Pascal’s triangle can be used to obtain row 6, and use the information to write the expansion of (x + y) 6 64411 10 5511 2015 66 11 Row 4 Row 5 Row 6 ++++ +++++ The Pattern?

7. Finding Binomial Coefficients Find the coefficient of x in the expansion of (x + 2). 10 The only term we need: 15 The coefficient of x 10

8. The Binomial Theorem For any positive integer n, where

9. Using the Theorem Expand We expand, with

10. Guided Practice Find the coefficient of the given term in the binomial expression. term, Coefficient: term, Coefficient:

11. Guided Practice Use the Binomial Theorem to expand

12. Guided Practice Find the fourth term of Fourth term:

13. Guided Practice Use the Binomial Theorem to expand

14. Factorial Identities

15. Basic Factorial Identities For any integer n > 1, n! = n(n – 1)! For any integer n > 0, (n + 1)! = (n + 1)n!

16. Guided Practice Prove that for all integers n > 2. Identity Identity

17. Guided Practice Prove that for all integers n > 2. Identity Definition of Factorial

18. Guided Practice Prove that for all integers n > 2. Cancellation Simplify:

19. Guided Practice Prove that for all integers n > 1. Part 1:

20. Guided Practice Prove that for all integers n > 1. Part 2:

21. Guided Practice Prove that for all integers n > 2.

22. Guided Practice Prove that

23. Guided Practice Prove that