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

1 Answer

(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.