binomialkey - McCombs Math 381 The Binomial Theorem and...

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McCombs Math 381 The Binomial Theorem and Pascal's Triangle The Binomial Theorem: Let x and y be variables, and let n be a nonnegative integer. x + y ( ) n = n 0 ! " # $ % x n + n 1 ! " # $ % x n 1 y + n 2 ! " # $ % x n 2 y 2 + ((( + n n 1 ! " # $ % xy n 1 + n n ! " # $ % y n = n j ! " # $ % j = 0 n ) x n j y j Interesting Facts: 1. For n ! 0 , n k ! " # $ % k = 0 n = 2 n 2. For n > 0 , ! 1 ( ) k n k " # $ % k = 0 n ( = 0 3. For n ! 0 , 2 k n k ! " # $ % k = 0 n = 3 n 4. Pascal's Identity: For n ! k > 0 , n + 1 k ! " # $ % = n k 1 ! " # $ % + n k ! " # $ % 5. For n ! r ! 0 , j r ! " # $ % j = r n = n + 1 r + 1 ! " # $ % Pascal's Triangle: 0 0 ! " # $ % 1 0 ! " # $ % 1 1 ! " # $ % 2 0 ! " # $ % 2 1 ! " # $ % 2 2 ! " # $ % 3 0 ! " # $ % 3 1 ! " # $ % 3 2 ! " # $ % 3 3 ! " # $ % 4 0 ! " # $ % 4 1 ! " # $ % 4 2 ! " # $ % 4 3 ! " # $ % 4 4 ! " # $ % . . . n 0 ! " # $ % n 1 ! " # $ % n 2 ! " # $ % n 3 ! " # $ % n 4 ! " # $ % ’ ’ ’ ’ n n ( 1 ! " # $ % n n ! " # $ % &
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McCombs Math 381 The Binomial Theorem and Pascal's Triangle Examples: 1. Use the binomial theorem to expand 2 a + b ( ) 4 . 2
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This note was uploaded on 06/16/2011 for the course MATH 381 taught by Professor Na during the Summer '11 term at University of North Carolina School of the Arts.

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binomialkey - McCombs Math 381 The Binomial Theorem and...

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