11-10 The Binomial Formula  Recall: (a + b) 2 = a 2 +2ab + b 2  For any binomial expansion: (a + b) n There are (n + 1) terms. Each term is n th degree. Each term has a coefficient.

The Binomal formula  The degree of each term: The exponent of a starts at n and decreases. The exponent of b starts at zero and increases.

The Binomial formula  Calculating the coefficient of each term: The next coefficient is predicted using the previous terms info:  (coefficient·the exponent of a) the term number The coefficients are symmetric.

Example: Expand (a + b) 4  (a + b) 4 =a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4  Term #  Notice: term # 3 The coefficient = 4(3)/2 The exponent of b = 3-1 = 2 The exponent of a = 4-2 = 2

Focus on the coefficient  For the 3rd term of (a + b) 4 : The coefficient = 4(3)/2 In factorial notation:

Summary  Term = n = exponent of the binomial r = exponent of b for a particular term

Practice  Find the 8 th term of (a + b) 14  8 th term = ___ a ? b 8-1  = ___ a 14-7 b 7  = ___ a 7 b 7  = Answer: 3432 a 7 b 7

Find the 5 th term of (x 2 – 2y) 11  5 th term = ______ (x 2 ) ? (2y) 5-1  = ______ (x 2 ) 11-4 (16y 4 )  = ______ (x 2 ) 7 (16y 4 )  = ______ (x 14 )(16y 4 )  = ______ 16x 14 y 4

Evaluating the coefficient  In factorial notation, the coefficient of the 5 th term of (x 2 – 2y) 11 is: = = 330, but remember the 16 (what 16?) = 330(16) = 5280

Putting this all together  the 5 th term of (x 2 – 2y) 11 is:  5280 x 14 y 4  Why is this term positive and not negative?

Let’s speed this up!  Find the term containing h 39 of (g-h) 64 Answer: - g 25 h 39 Simplified: - g 25 h 39

Last Example  Find the 14 th term of (x – y) 16  Answer: - x 3 y 13  = - x 3 y 13

Homework:  P , Q1-10; #1 – 47 EOO