(x^3-x^2+1) / (x^4-x^3) = (x^3-x^2)/ (x^4-x^3) + 1/(x^4-x^3)x3−x2+1x4−x3=x3−x2x4−x3+1x4−x3
=(x-1)/(x^2-x) + 1/(x^3(x-1))=x−1x2−x+1x3(x−1)
=(x-1)/(x(x-1)) + 1/(x^3(x-1))=x−1x(x−1)+1x3(x−1)
=1/x + 1/(x^3(x-1))=1x+1x3(x−1)
now focus on 1/(x^3(x-1))1x3(x−1)
1/(x^3(x-1)) = A/x^3+B/x^2+C/x+D/(x-1)1x3(x−1)=Ax3+Bx2+Cx+Dx−1
Multiply both side by x^3(x-1)x3(x−1)
1 = A(x-1)+Bx(x-1)+Cx^2(x-1)+Dx^31=A(x−1)+Bx(x−1)+Cx2(x−1)+Dx3
1=Ax-A+Bx^2-Bx+Cx^3-Cx^2+Dx^31=Ax−A+Bx2−Bx+Cx3−Cx2+Dx3
1=x^3(C+D)+x^2(B-C)+x(A-B)-A1=x3(C+D)+x2(B−C)+x(A−B)−A
C+D = 0C+D=0
B-C = 0B−C=0
A-B = 0A−B=0
A = -1A=−1
Just by looking we have
A = - 1A=−1
B = -1B=−1
C = -1C=−1
D = 1D=1
So
(x^3-x^2+1) / (x^4-x^3) = -1/x^3-1/x^2+1/(x-1)x3−x2+1x4−x3=−1x3−1x2+1x−1
