Posted by **Robert** on Sunday, December 4, 2011 at 9:54pm.

Write the partial fraction decomposition of the rational expression.

6x^2+1/x^2(x-1)^2

- precalc -
**Reiny**, Sunday, December 4, 2011 at 10:39pm
Because of the duplication of factors we could have

A/x + B/x^2 + C/(x-1) + D/(x-1)^2 = (6x^2 + 1)/(x^2(x-1)^2)

multiply by x^2(x-1)^1

A(x-1)^2 + Bx(x-1)^2 + Cx^2(x-1) + Dx^2 = 6x^2 + 1

let x=0 --> A = 1

let x=1 --> D = 7

then (x-1)^2 + Bx(x-1)^2 + Cx^2(x-1) + 7x^2 = 6x^2 + 1

let x=2 --> 1 + 2B + 4C + 28 = 24+1

or B + 2C = -2 (#1)

let x = -1 --> 4 - 4B - 2C + 7 = 7

or 2B + C = 2 (#2)

2 (#1) - #2

3C = -6

C = -2

then B = 2 form #1

so (6x^2 + 1)/(x^2(x-1)^2)

= 1/x^2 + 2/x - 2/(x-1) + 7/(x-1)^2

- precalc -
**Steve**, Monday, December 5, 2011 at 11:25am
Looks like it should have said

A/x^2 + B/x + C/(x-1) + D/(x-1)^2 = (6x^2 + 1)/(x^2(x-1)^2)

but the solution from there on is correct.

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