2.4 Use the Binomial Theorem Test: Friday

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

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.

Vocabulary Binomial Expansion (a + b) 0 = 1 (a + b) 1 = 1a + 1b (a + b) 2 = 1a 2 + 2ab + 1b 2 (a + b) 3 = 1a 3 + 3a 2 b + 3ab 2 + 1b 3 (a + b) 4 = 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + 1b 4

Vocabulary  Binominal Expansion (a + b) 3 1a 3 + 3a 2 b + 3ab 2 + 1b 3 *as the a exponents decrease, the b exponents increase Where do the coefficients (1, 3, 3, 1) come from?

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

Example:  Use the forth row of Pascal’s triangle to find the numbers in the fifth row of Pascal’s triangle

Example:  Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x + 2) 3 Binomial Theorem: (a + b) 3 = a 3 + a 2 b + ab 2 + b 3 Pascal’s Triangle: row Together: 1a 3 + 3a 2 b + 3ab 2 + 1b 3

Example Continued: (x + 2) 3 ab 1a 3 + 3a 2 b + 3ab 2 + 1b 3 1(x) 3 + 3(x) 2 (2) + 3(x)(2) 2 + 1(2) 3 X 3 + 6x x + 8

You Try:  Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x + 1) 4  Solution: a = x, b = 1 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + 1b 4 x 4 + 4x 3 + 6x 2 + 4x + 1

Example:  Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x – 3) 4 watch out for the negative! 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + 1b 4 1(x) 4 + 4(x) 3 (-3) + 6(x) 2 (-3) 2 + 4(x)(-3) 3 + 1(-3) 4 x 4 – 12x x 2 – 108x + 81 ab

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