1. Objective: To use Pascal’s Triangle and to explore the Binomial Theorem.

2.  FOIL: (a+b) 2 = (a+b)(a+b) = a 2 + 2ab + b 2  Distributive Property: (a+b)(a+b) = a(a+b) + b(a+b) = a 2 + 2ab + b 2  We are going to learn a shortcut method for expanding a binomial.

3.  (a+b) 2 = a 2 + 2ab + b 2  (a + b) 3 = (a + b)(a + b)(a + b) = a 3 + 3a 2 b + 3ab 2 + b 3 In the first case, coefficients are 1, 2, 1. In the second case, coefficients are 1, 3, 3, 1.

4.  Pascal’s Triangle is a triangular array of numbers with a border made of 1’s.  The first row has 1 element, the second row has 2 elements,…the nth row has n elements.  This triangle is used to find coefficients of a binomial expansion.

5. 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Notice anything?

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7. 1. Write the row of Pascal’s Triangle whose second element corresponds to the power of the binomial. 2. Write decreasing powers of a from left to right, beginning with a 3. 3. Write increasing powers of b from left to right, beginning with b 3.

8.  Use Pascal’s Triangle to expand. A) (a+b) 8

9. This is the binomial coefficient in the previous expansion of binomials It may also be considered as the combination of n objects taken r at a time.

10.  Check Point 1: Evaluate

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12.  The binomial theorem provides a means to expand a binomial expression.  It provides a “shortcut” to taking an expression, such as (a+b) and raising it to a power (n) without having to continue to multiply binomials out.

14.  Check Point 3: Expand (x – 2y) 5.

15. Copyright © 2005 Pearson Education, Inc.  Finding the kth Term The k th term of (a + b) n = Example: Find the 7 th term in the expansion (x 2  2y ) 11.

16. First, we note that k=7. Thus, k = 6, a = x 2, b =  2y, and n = 11. Then the 7 th term of the expansion is

17.  Check Point 4: Find the fifth term of (2x + y) 9.