Setting up partial fraction decomposition Give the appropriate farm of the partial fraction decomposition for the following functions.
Problem 7ESetting up partial fraction decomposition Give the appropriate farm of the partial fraction decomposition for the following functions. Answer;Step-1Proper fraction definition ; In a rational fraction , if the degree of f(x) < the degree of g(x) , then the rational fraction is called a proper fraction. The sum of two proper fractions is a proper fraction. Example; Partial fractions Depending upon the nature of factors of Denominator ;1. When the denominator has non-repeated linear factors; A non - repeated linear factor (x-a) of denominator corresponds a partial fraction of the form , where A is a constant to be determined’ If g(x) = (x-a)(x-b)(x-c)............(x-n), then we assume that = ++ +...............+Where A, B, C,............N are constants which can be determined by equating the numerator of L.H.Sto the numerator of R.H.S , and substituting x = a,b ,c ….n.Step-2 The given fraction is ; The given fraction is form , and f(x) < g(x). Therefore , the given fraction is a proper fraction , and the denominator has non- repeated linear factors. Thus , from the above step the given fraction can be written as; = = = , since (a+b)(a-b) = = Therefore , = = +…………….(1) = Thus , = = Equating the numerator of L.H.S to the numerator of R.H.S Then , ( x) = A(x+4) +B(x-4) X = (A+B)x+ (A-B) A+B = 1 , and (A-B) = 0.So , A = B Solving A+B=1, A- B = 0 , then B = , and A+B= 1 A = 1 -B = 1 -= A = ………..(2) Therefore, from(1), (2) = = = + , since from(1) = [+] Therefore , = [+]