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Exploring the Binomial Theorem: Coefficients and Patterns in Expansion

Dive into the fascinating world of the Binomial Theorem, where we unravel the coefficients found in each term of binomial expansions. You'll discover how the exponents of terms sum to "n" and explore patterns among the coefficients. Through examples such as (x + y)^2, (x + y)^3, and (x + y)^6, we investigate how each coefficient follows a distinct sequence, often represented by Pascal's Triangle. Learn to find coefficients using formulas and understand their significance in combinatorial mathematics.

hermione-hodge
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Exploring the Binomial Theorem: Coefficients and Patterns in Expansion

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  1. The Binomial Theorem

  2. What about the coefficients of each term? Is there a pattern there? Find the patterns: 1 (x + y)0 (x + y)0 x1 + y1 x + y (x + y)1 (x + y)1 x2 + 2xy + y2 (x + y)2 (x + y)2 (x + y)3 (x + y)3 x3 y0 + 3x2 y1 + 3x1 y2+ x0 y3 x3 + 3x2 y1 + 3x1 y2+ y3 x3 y0 + 3x2 y1 + 3x1 y2+ x0 y3 x3 + 3x2 y + 3xy2+ y3 (x + y)4 (x + y)4 x4 + 4x3 y + 6x2 y2 +4xy3 + y4 Notice how the exponents of each term sum to “ n “. (x + y)6 x6 + 6x5 y + 15x4 y2 +20x3 y3 + 15x2 y4 + 6x1 y5 + y6

  3. What about the coefficients of each term? Is there a pattern there? 0 1 1 1 1 1 2 1 2 1 1 3 3 3 1 4 6 4 1 4 10 5 1 1 5 10 5

  4. Finding the Coefficient of a Binomial using a Formula. n =

  5. Finding the Coefficient of a Binomial using a Formula. 6 = = x 3 = x x

  6. Finding the Coefficient of a Binomial using a Formula. =

  7. Finding the Coefficient of a Binomial using a Formula. 5 = 2 x = x x

  8. What about the coefficients of each term? Is there a pattern there? 0 1 1 1 1 1 2 1 2 1 1 3 3 3 1 4 6 4 1 4 10 5 1 1 5 10 5 5 5 5 5 5 5

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