1078 CHAPTER 11 Further Topics in Algebra EXAMPLE 4 Applying the BinomialTheorem Expand a 3 m2 - 22mb 4 . (Assume m70.) SOLUTION Apply the binomial theorem. a 3 m2 - 22mb 4 = a 3 m2b 4 + a 4 1b a 3 m2b 3A -22m B + a 4 2b a 3 m2b 2A -22m B 2 + a 4 3b a 3 m2bA -22m B 3 + A -22m B4 = 81 m8 + 4 a 27 m6b 1-2m1/22 + 6 a 9 m4b 14m2 + 4 a 3 m2b 1-8m3/22 + 16m2 2 m= m1/2 = 81 m8 - 216 m11/2 + 216 m3 - 96 m1/2 + 16m2 S Now Try Exercise 45. k th Term of a Binomial Expansion Earlier in this section, we wrote the binomial theorem in summation notation as a n r=0A n r B x n-ryr, which gives the form of each term. We can use this form to write any particular term of a binomial expansion without writing out the entire expansion. kthTerm of the Binomial Expansion The kth term of the binomial expansion of 1x + y2n, where n Ú k - 1, is given as follows. a n k −1b xn−1k−12 yk−1 To find the kth term of the binomial expansion, use the following steps. Step 1 Find k - 1. This is the exponent on the second term of the binomial. Step 2 Subtract the exponent found in Step 1 from n to obtain the exponent on the first term of the binomial. Step 3 Determine the coefficient by using the exponents found in the first two steps and n. EXAMPLE 5 Finding a ParticularTerm of a Binomial Expansion Find the seventh term of the binominal expansion of 1a + 2b210. SOLUTION In the seventh term, 2b has an exponent of 7 - 1, or 6, while a has an exponent of 10 - 6, or 4. a10 6 b a412b26 Seventh term of the binomial expansion = 210a4164b62 Evaluate and apply the power rule for exponents. = 13,440a4b6 Multiply. S Now Try Exercise 49.
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