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13.5 The Binomial Theorem

Dive into the Binomial Theorem, a vital mathematical principle used to expand expressions like (x + y)⁴ or (2x - 5y)⁷. Explore Pascal's Triangle, which provides coefficients for binomial expansions by combining diagonal numbers. Discover how to derive each term using factorials, identify specific terms in expansions, and calculate missing exponents. Through practical examples and exercises, sharpen your skills in expanding binomials and understanding the connections within Pascal's Triangle.

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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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