Stitz-Zeager_College_Algebra_e-book

Example 1024 suppose is an acute angle with cos 5 13

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Unformatted text preview: j !(n − j + 1)! = n! (j + (n − j + 1)) j !(n − j + 1)! = (n + 1)n! j !(n + 1 − j ))! = (n + 1)! j !((n + 1) − j ))! n+1 j We are now in position to state and prove the Binomial Theorem where we see that binomial coeﬃcients are just that - coeﬃcients in the binomial expansion. Theorem 9.4. Binomial Theorem: For nonzero real numbers a and b and natural numbers n, = n n (a + b) = j =0 n n−j j ab j To get a feel of what this theorem is saying and how it really isn’t as hard to remember as it may ﬁrst appear, let’s consider the speciﬁc case of n = 4. According to the theorem, we have 4 (a + b)4 = j =0 4 4−j j ab j = 4 4−0 0 4 4−1 1 4 4−2 2 4 4−3 3 4 4−4 4 a b+ a b+ a b+ a b+ ab 0 1 2 3 4 = 44 43 4 22 4 44 a+ a b+ ab + ab3 + b 0 1 2 3 4 9.4 The Binomial Theorem 585 We forgo the simpliﬁcation of the coeﬃcients in order to note the pattern in the expansion. First note that in each term, the total of the exponents is 4 which matched the exponent of the binomial (a + b)4 . The exponent on a begins at 4 and decreases by one as we move from one term to t...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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