1077 11.4 The Binomial Theorem = x9 + 9! 1! 8! x8y + 9! 2! 7! x7y2 + 9! 3! 6! x6y3 + 9! 4! 5! x5y4 + 9! 5! 4! x4y5 + 9! 6! 3! x3y6 + 9! 7! 2! x2y7 + 9! 8! 1! xy8 + y9 = x9 + 9x8y + 36x7y2 + 84x6y3 + 126x5y4 + 126x4y5 + 84x3y6 + 36x2y7 + 9xy8 + y9 Evaluate each binomial coefficient. S Now Try Exercise 31. aa - b 2b 5 = a5 + a 5 1b a4 a- b 2b + a 5 2b a3 a- b 2b 2 + a 5 3b a2 a- b 2b 3 + a 5 4b a a- b 2b 4 + a- b 2b 5 = a5 + 5a4 a- b 2b + 10a3 a- b 2b 2 + 10a2 a- b 2b 3 + 5a a- b 2b 4 + a- b 2b 5 = a5 - 5 2 a4b + 5 2 a3b2 - 5 4 a2b3 + 5 16 ab4 - 1 32 b5 In this expansion, the signs of the terms alternate (as shown in color) because y = - b 2 has a negative sign. S Now Try Exercise 43. EXAMPLE 2 Applying the BinomialTheorem Write the binomial expansion of 1x + y29. SOLUTION Apply the binomial theorem. 1x + y29 = x9 + a 9 1bx8y + a 9 2bx7y2 + a 9 3bx6y3 + a 9 4bx5y4 + a 9 5bx4y5 + a 9 6bx3y6 + a 9 7bx2y7 + a 9 8bxy8 + y9 EXAMPLE 3 Applying the BinomialTheorem Expand aa - b 2b 5 . SOLUTION Write the binomial as follows. aa - b 2b 5 = aa + a- b 2bb 5 Now apply the binomial theorem with x = a, y = - b 2 , and n = 5. NOTE As Example 3 illustrates, an expansion of the difference of two terms ( for example, an expansion of 1x −y2n for n #2) has alternating signs.
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