Stitz-Zeager_College_Algebra_e-book

In each instance an initial value of the sequence is

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Unformatted text preview: B1 , B2 , . . . Bm and C1 , C2 , . . . Cm . a In other words, R(x) is a proper rational function. The proof of Theorem 8.10 is best left to a course in Abstract Algebra. Notice that the theorem provides for the general case, so we need to use subscripts, A1 , A2 , etc., to denote different unknown coefficients as opposed to the usual convention of A, B , etc.. The stress on multiplicities is to help us correctly group factors in the denominator. For example, consider the rational function (x2 3x − 1 − 1) (2 − x − x2 ) Factoring the denominator to find the zeros, we get (x + 1)(x − 1)(1 − x)(2 + x). We find x = −1 and x = −2 are zeros of multiplicity one but that x = 1 is a zero of multiplicity two due to the two different factors (x − 1) and (1 − x). One way to handle this is to note that (1 − x) = −(x − 1) so 3x − 1 3x − 1 1 − 3x = = (x + 1)(x − 1)(1 − x)(2 + x) −(x − 1)2 (x + 1)(x + 2) (x − 1)2 (x + 1)(x + 2) from which we proceed with the partial fraction decomposit...
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