1. 13.5 The Binomial Theorem

2. There are several theorems and strategies that allow us to expand binomials raised to powers such as (x + y) 4 or (2x – 5y) 7. One of these is Pascal’s Triangle 1 1 121121 1 3 3 1 1464114641 1 5 10 10 5 1 1615201561 each row starts & ends with a 1 to get each term, you add the two numbers diagonally above that spot *we can continue like this forever! your turn: again:

3. Another is the idea of factorials *remember it can also be written as Ex 1) Find the lie and fix it! A)B) C) No  4368 There is a connection between the numbers in Pascal’s Triangle and Take for instance Row 4  14641 notice that This helps us both expand binomials as well as find a particular term of an expansion without expanding the whole thing.

4. How to expand a binomial: (a + b) n (1)Coefficients: use Pascal’s Triangle or (2)Powers of each variable: the powers on the first term descend fromn …. 0 the powers on the second term ascend from0 …. n Ex 2) Using Pascal’s Triangle, expand (x + y) 4 1 x 4 y 0 + 4 x 3 y 1 + 6 x 2 y 2 + 4 x 1 y 3 + 1 x 0 y 4 x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4

5. Ex 3) Expand Ex 4) Find the coefficient of the indicated term & identify the missing exponent. x ? y 9 ; (x + y) 11 x2y9x2y9 *each set of powers adds to the total of 11 ? + 9 = 11 ? = 2 this is simply

6. Ex 5) Find the term involving the specified variable. b 6 in (a – b) 14 Ex 6) Find the indicated term of the expansion. a) the third term of (a + 5b) 4 b) the fourth term of (2a – 6b) 11 *the third term would have a 2 (count down) *fourth term a 11 a 10 a 9 a 8

7. Homework #1305 Pg 712 #1–9 odd, 13, 17, 19, 21, 23–27, 32, 33