SECTION 12.6 The Binomial Theorem 901 This is why it was necessary to introduce the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j ; the numbers ( ) − n j n j ! ! ! are the numerical coefficients in the expansion of ( ) +x a . n Because of this, the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j is called a binomial coefficient. Expanding a Binomial Use the Binomial Theorem to expand ( ) +x 2 . 5 Solution EXAMPLE 2 In the Binomial Theorem, let = a 2 and = n 5. Then ( ) + = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = + + + + + x x x x x x x x x x x x x x x x 2 5 0 5 1 2 5 2 2 5 3 2 5 4 2 5 5 2 1 5 2 10 4 10 8 5 16 1 32 10 40 80 80 32 5 5 4 2 3 3 2 4 5 5 4 3 2 5 4 3 2 ↑ Use equation (2). ↑ Use row =n 5 of the Pascal triangle or definition (1) for ⎛ ⎝ ⎜⎜ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎟ n j . Expanding a Binomial Expand ( ) −y2 3 4 using the Binomial Theorem. Solution EXAMPLE 3 First, rewrite the expression ( ) −y2 3 4 as [ ] ( ) + − y2 3 . 4 Now use the Binomial Theorem with = = n x y 4, 2 , and = − a 3. [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + − = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − = ⋅ + − + ⋅ ⋅ + − + ⋅ = − + − + y y y y y y y y y y y y y 2 3 4 0 2 4 1 3 2 4 2 3 2 4 3 3 2 4 4 3 1 16 4 3 8 6 9 4 4 27 2 1 81 16 96 216 216 81 4 4 3 2 2 3 4 4 3 2 4 3 2 In this expansion, note that the signs alternate because = − < a 3 0. ↑ Use row =n 4 of the Pascal triangle or definition (1) for ⎛ ⎝ ⎜⎜ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎟ n j . Now Work PROBLEM 21 Finding a Particular Coefficient in a Binomial Expansion Find the coefficient of y8 in the expansion of ( ) +y2 3 . 10 Solution EXAMPLE 4 Expand ( ) +y2 3 10 using the Binomial Theorem. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ y y y y y y y 2 3 10 0 2 10 1 2 3 10 2 2 3 10 3 2 3 10 4 2 3 10 9 2 3 10 10 3 10 10 9 1 8 2 7 3 6 4 9 10 From the third term in the expansion, the coefficient of y8 is ( ) ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = 10 2 2 3 10! 2! 8! 2 9 10 9 8! 2 8! 2 9 103,680 8 2 8 8
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