# How do you write the partial fraction decomposition of the rational expression #(x+1)/( (x^2 )*(x-1) )#?

##### 1 Answer

#### Explanation:

The decomposition that should be set up from this problem is

#(x+1)/(x^2(x-1))=A/x+B/x^2+C/(x-1)#

Note that since

From here, we can get a common denominator of

#(x+1)/(x^2(x-1))=(Ax(x-1))/(x^2(x-1))+(B(x-1))/(x^2(x-1))+(Cx^2)/(x^2(x-1))#

The denominators are equal, so they can be removed, giving us the equation

#x+1=Ax(x-1)+B(x-1)+Cx^2#

Set

#1+1=A(1)(0)+B(0)+C(1)#

#ul(C=2#

Set

#0+1=A(0)(-1)+B(-1)+C(0)#

#ul(B=-1#

Now that we know the values of

#2+1=A(2)(1)+(-1)(1)+2(4)#

#3=2A+7#

#ul(A=-2#

This leaves us with the decomposition of

#(x+1)/(x^2(x-1))=-2/x-1/x^2+2/(x-1)#