How do you write the partial fraction decomposition of the rational expression #(6x^2+1)/(x^2(x-1)^2)#?

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Answer 1 · 2017-10-08T08:54:56

#(6x^2 + 1)/[x^2(x - 1)^2]-= (Ax + B)/x^2 + C/(x - 1) + D/(x - 1)^2#

so, #6x^2 + 1 = (Ax + B)(x - 1)^2 + Cx^2(x - 1) + Dx^2#

Setting x = 1 we have:

#7 = D#

With x = 0 we have:

#1 = B#

Withe x = -1 we get:

#7 = 4(B - A) - 2C + D#

so, #7 = 4(1 - A) - 2C + 7#

Hence, #2A + C =2#....(1)

With x = 2 we get:

#25 = 2A + B + 4C + 4D#

so, 25 = 2A + 1 + 4C + 28#

Hence, #A + 2C = -2#...(2)

Now, #2(2) - (1)# gives:

#3C = -6#

so, #C = -2#

i.e. #A = 2#

Therefore

#(6x^2 + 1)/[x^2(x - 1)^2]-= (2x + 1)/x^2 - 2/(x - 1) + 7/(x - 1)^2#

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