1. Algebra 2 CC 1.3 Apply the Binomial Expansion Theorem Recall: A binomial takes the form; (a+b) Complete the table by expanding each power of a binomial. (a+b) 0 (a+b) 1 (a+b) 2 (a+b) 3 (a+b) 4 (a+b) 7

2. Pascal's Triangle We note that the coefficients (the numbers in front of each term) follow a pattern. 1 1 1 2 1 1 3 3 1 1 4 6 4 1 You can use this pattern to form the coefficients, rather than multiply everything out. (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 + 4ab 3 + b 4

3. The Binomial Theorem We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases. Properties of the Binomial Expansion (a + b) n There are n+1 terms. The first term is a n and the final term is b n. Progressing from the first term to the last, the exponent of a decreases by 1 from term to term while the exponent of b increases by 1. In addition, the sum of the exponents of a and b in each term is n. (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 + 4ab 3 + b 4

4. Use Pascal’s Triangle and the Binomial Expansion Theorem to simplify. (a+b) 7 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

5. Use Pascal’s Triangle and the Binomial Expansion Theorem to simplify. (x - y) 5 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

6. Use Pascal’s Triangle and the Binomial Expansion Theorem to simplify. (2x – 3y) 4 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

7. Find the 7 th term of the expansion (2x – 3y) 9 Assignment: worksheet Alg 2 cc1.3 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