CVPL1

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Find the partial fraction decomposition of {\frac  {4z+4}{z(z-1)(z-2)^{2}}}\,.

{\frac  {4z+4}{z(z-1)(z-2)^{2}}}={\frac  {A}{z}}+{\frac  {B}{z-1}}+{\frac  {C}{z-2}}+{\frac  {D}{(z-2)^{2}}}\,

4z+4=A(z-1)(z-2)^{2}+Bz(z-2)^{2}+Cz(z-1)(z-2)+Dz(z-1)\,

Plug in z=0\, to get A=-1\,

Plug in z=1\, to get B=8\,

Plug in z=2\, to get D=6\,

Differentiate both sides of the equation once with respect to z\, and plug in z=2\, to get C=-7\,.

Finally

{\frac  {4z+4}{z(z-1)(z-2)^{2}}}={\frac  {-1}{z}}+{\frac  {8}{z-1}}-{\frac  {7}{z-2}}+{\frac  {6}{(z-2)^{2}}}\,


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