2.4 Use the Binomial Theorem

1. 2.4Use the Binomial Theorem 2.1-2.5 Test: Friday

2. Think about this… • Expand (x + y)12 Would you want to multiply (x +y) 12 times?!?!?!

3. Vocabulary • Binomial Theorem and Pascal’s Triangle • The numbers in Pascal’s triangle can be used to find the coefficients in binomial expansions (a + b)n where n is a positive integer.

4. Vocabulary Binomial Expansion (a + b)0 = 1 (a + b)1 = 1a + 1b (a + b)2 = 1a2 + 2ab + 1b2 (a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3 (a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4

5. Vocabulary • Binominal Expansion (a + b)3 1a3 + 3a2b + 3ab2 + 1b3 *as the a exponents decrease, the b exponents increase Where do the coefficients (1, 3, 3, 1) come from?

6. Vocabulary Pascal’s Triangle • 1 • 1 • 1 2 1 • 1 3 3 1 • 1 4 6 4 1 n = 0 (0th row) n = 1 (1st row) n = 2 (2nd row) n = 3 (3rd row) n = 4 (4th row) The first and last numbers in each row are 1. Beginning with the 2nd row, every other number is formed by adding the two numbers immediately above the number

7. Example: • Use the forth row of Pascal’s triangle to find the numbers in the fifth row of Pascal’s triangle. 1 4 6 4 1 1 5 10 10 5 1

8. Example: Binomial Theorem: (a + b)3 = a3 + a2b+ ab2 + b3 Pascal’s Triangle: row 3 1 3 3 1 Together: 1a3 + 3a2b+ 3ab2 + 1b3 • Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x + 2)3

9. Example Continued: (x + 2)3 a b 1a3 + 3a2b + 3ab2 + 1b3 1(x)3 + 3(x)2(2) + 3(x)(2)2 + 1(2)3 X3 + 6x2 + 12x + 8

10. You Try: • Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x + 1)4 • Solution: a = x, b = 1 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4 x4 + 4x3 + 6x2 + 4x + 1

11. Example: • Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x – 3)4 • watch out for the negative! 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4 1(x)4 + 4(x)3(-3) + 6(x)2(-3)2 + 4(x)(-3)3 + 1(-3)4 x4 – 12x3 + 54x2 – 108x + 81 a b

12. Homework: • p. 75 # 1-13odd • Due tomorrow