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

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Answer 1 · 2016-02-01T08:55:31

Explanation is given below

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

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

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

Equating the numerators

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

To make this easy for us, let us start by taking #x=0# and plug in the value, we get.

#0+1=A(0)(0-2)+B(0-2)+C(0)^2#
#1=0-2B+0#

#-2B=1#
#B=-1/2#

Note #x=0# helped us eliminate #A# and #C# now let us take #x=2#
#2+1 = A(2)(2-2)+B(2-2)+C(2^2)#
#3=A(2)(0)+B(0)+4C#
#3=0+0+4C#

#4C=3#
#C=3/4#

We have to find #A# Let us plug in x=1
#1+1=A(1)(1-2)+B(1-2)+C(1^2)#
#2=A(1)(-1)-B+C#

#-A-B+C = 2#
#-A-(-1/2)+3/4=2#

#-A+1/2+3/4=2#

#-A+2/4+3/4=2#
#-A+5/4=2#

#-A=2-5/4#

#-A=(8-5)/4#

#-A=3/4#

#A=-3/4#

#(x+1)/(x^2(x-2)) = -3/(4x)-1/(2x^2) + 3/(4(x-2))#