Question

How do you express `(x+1)/( (x^2 )*(x-1) )` in partial fractions?

Answer 1

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

Explanation:

From the given `(x+1)/(x^2(x-1))`, we set up the equations with the needed unknown constants represented by letters

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

The Least Common Denominator is `x^2(x-1)`.

Simplify

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

`(x+1)/(x^2(x-1))=(A(x-1)+B(x^2-x)+Cx^2)/(x^2(x-1))`

Expand the numerator of the right side of the equation

`(x+1)/(x^2(x-1))=(Ax-A+Bx^2-Bx+Cx^2)/(x^2(x-1))`

Rearrange from highest degree to the lowest degree the terms of the numerator at the right side

`(x+1)/(x^2(x-1))=(Bx^2+Cx^2+Ax-Bx-A)/(x^2(x-1))`

Now let us fix the numerators on both sides so that the numerical coefficients are matched.

`(0*x^2+1*x+1*x^0)/(x^2(x-1))=((B+C)x^2+(A-B)x-A*x^0)/(x^2(x-1)`

We can now have the equations to solve for A, B, C

`B+C=0`
`A-B=1`
`-A=1`

Simultaneous solution results to
`A=-1`
`B=-2`
`C=2`

We now have the final equivalent
`(x+1)/(x^2(x-1))=-1/x^2-2/x+2/(x-1)`

God bless....I hope the explanation is useful.