1. Lesson 6.8A: The Binomial Theorem OBJECTIVES:  To evaluate a binomial coefficient  To expand a binomial raised to a power

2. VOCABULARY n! (read: “n factorial”) is ____________ ______________________________ ______________________________. the product of all positive integers less than or equal to n. n! = n (n - 1)(n - 2)(n - 3)...(1)

3. 0!=1 1! =_______________________ 2! =_______________________ 3! =_______________________ 4! =_______________________ 5! =_______________________ 6! =_______________________ 7! =_______________________ 8! =_______________________ 9! =_______________________ Evaluate the factorials. 1 0! = 1 2 1 0! = 2 3 2 1 0! = 6 4 3 2 1 0! = 24

4. BINOMIAL COEFFICIENTS When raising a binomial to a larger power, a more efficient way to determine the coefficients in a binomial expansion is to write them in terms of factorials.

5. Definition of a Binomial Coefficient Definition of a Binomial Coefficient, For nonnegative integers n and r, with n ≥ r, the expression (read “n above r”) is called a binomial coefficient and is defined by =

6. EXAMPLE 1 Evaluating Binomial Coefficients In each case, apply the definition of the binomial coefficient. a. = _________ =____ Evaluate: a. b. c. d. NOTE: The symbol n C r is often used in place of to denote binomial coefficients. a. 15 b. 84 c. 1 d. 1

7. THE BINOMIAL THEOREM A Formula for Expanding Binomials: The Binomial Theorem For any positive integer n, (a + b) n = a n + a n-1 b + a n-2 b 2 + a n-3 b 3 +…+ b n

8. Using the Binomial Theorem Expand: (x + 2) 4 Note: a = x; b = 2, and n = 4 _______ + _______ + _______ + _______ + _______ x 4 + 8x 3 + 24x 2 + 32x + 16 Solution (x +2) 4 =

9. Technology Graphing calculators can compute binomial coefficients. For example to find, many calculators require the sequence 6 nCr 2 ENTER. Use your calculator to verify the other evaluations in example 1.

10. FINAL CHECKS FOR UNDERSTANDING  Describe the pattern on the exponents on a in the expansion (a + b) n.  Describe the pattern on the exponents on b in the expansion of (a + b) n.  What is true about the sum of the exponents on a and b in any term of the expansion of (a + b) n ?  How do you determine how many terms there are in a binomial expansion?

11. FINAL CHECKS FOR UNDERSTANDING  What is Pascal’s triangle? How do you find the numbers in any row of the triangle?  Explain how to evaluate. Provide an example with your explanation.  Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.  Are situations in which it is easier to use Pascal’s triangle than binomial coefficients? Describe these situations.

12. Homework Assignment Binomial Theorem WS 1-7 odd (Verify your answers using your ) 9, 10, 19, 23, 30, 60* (*calculator).