From the given (x+1)/(x^2(x-1))x+1x2(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)x+1x2(x−1)=Ax2+Bx+Cx−1
The Least Common Denominator is x^2(x-1)x2(x−1).
Simplify
(x+1)/(x^2(x-1))=A/x^2+B/x+C/(x-1)x+1x2(x−1)=Ax2+Bx+Cx−1
(x+1)/(x^2(x-1))=(A(x-1)+B(x^2-x)+Cx^2)/(x^2(x-1))x+1x2(x−1)=A(x−1)+B(x2−x)+Cx2x2(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))x+1x2(x−1)=Ax−A+Bx2−Bx+Cx2x2(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))x+1x2(x−1)=Bx2+Cx2+Ax−Bx−Ax2(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)0⋅x2+1⋅x+1⋅x0x2(x−1)=(B+C)x2+(A−B)x−A⋅x0x2(x−1)
We can now have the equations to solve for A, B, C
B+C=0B+C=0
A-B=1A−B=1
-A=1−A=1
Simultaneous solution results to
A=-1A=−1
B=-2B=−2
C=2C=2
We now have the final equivalent
(x+1)/(x^2(x-1))=-1/x^2-2/x+2/(x-1)x+1x2(x−1)=−1x2−2x+2x−1
God bless....I hope the explanation is useful.
