Stitz-Zeager_College_Algebra_e-book

From this we determine is a quadrant iii or quadrant

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Unformatted text preview: this and the fact that k+1 0 = 1 and k+1 k+1 k k+1−j j a b+ j k j =1 k k + j j−1 k + 1 k+1−j j a b j = 1, we get k ak+1−j bj j−1 ak+1−j bj 9.4 The Binomial Theorem 587 k k + 1 k+1−j j a b + bk+1 j (a + b)k+1 = ak+1 + j =1 = k + 1 k+1 0 a b+ 0 k+1 = j =0 k k + 1 k+1−j j k + 1 0 k+1 a b+ ab j k+1 j =1 k + 1 (k+1)−j j a b j which shows that P (k + 1) is true. Hence, by induction, we have established that the Binomial Theorem holds for all natural numbers n. Example 9.4.2. Use the Binomial Theorem to find the following. 1. (x − 2)4 2. 2.13 3. The term containing x3 in the expansion (2x + y )5 Solution. 1. Since (x − 2)4 = (x + (−2))4 , we identify a = x, b = −2 and n = 4 and obtain 4 (x − 2)4 = j =0 = 4 4−j x (−2)j j 4 4−0 4 4−1 4 4−2 4 4−3 4 4−4 x (−2)0 + x (−2)1 + x (−2)2 + x (−2)3 + x (−2)4 0 1 2 3 4 = x4 − 8x3 + 24x2 − 32x + 16 2. At first this problem seem misplaced, but we can write 2.13 = (2 + 0.1)3 . Identifying a = 2, 1 b = 0.1 = 10...
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