Sunday, August 15, 2010

`(2x - 3)/(x - 1)^2` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

`(2x-3)/(x-1)^2`


Let`(2x-3)/(x-1)^2=A/(x-1)+B/(x-1)^2`


`(2x-3)/(x-1)^2=(A(x-1)+B)/(x-1)^2`


`(2x-3)/(x-1)^2=(Ax-A+B)/(x-1)^2`


`:.(2x-3)=Ax-A+B`


comparing the coefficients of the like terms,


`A=2`


`-A+B=-3`


Plug in the value of A in the above equation,


`-2+B=-3`


`B=-3+2`


`B=-1`


`:.(2x-3)/(x-1)^2=2/(x-1)-1/(x-1)^2`


Now let's check the above result,


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


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


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


`=(2x-3)/(x-1)^2`


`= LHS`


Hence it is verified.

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