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13.5 The Binomial Theorem

13.5 The Binomial Theorem. There are several theorems and strategies that allow us to expand binomials raised to powers such as ( x + y ) 4 or (2 x – 5 y ) 7 . One of these is Pascal’s Triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

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13.5 The Binomial Theorem

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  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’sTriangle 1 1 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 your turn: again: • 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!

  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  1 4 6 4 1 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 Coefficients: use Pascal’s Triangle or Powers of each variable: the powers on the first term descend from n …. 0 the powers on the second term ascend from 0 …. n Ex 2) Using Pascal’s Triangle, expand (x + y)4 1 x4y0 + 4 x3y1 + 6 x2y2 + 4 x1y3 + 1 x0y4 x4 + 4x3y + 6x2y2 + 4xy3 + y4

  5. Ex 3) Expand Ex 4) Find the coefficient of the indicated term & identify the missing exponent. x?y9 ; (x + y)11 x2y9 *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. b6 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 a2 (count down) *fourth term a11a10a9a8

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

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