How do you write the partial fraction decomposition of the rational expression # [(3x)/(x^2-6x+9)]#?

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Answer 1 · 2016-02-20T20:49:57

#3/(x-3)+9/(x-3)^2#

Start off by factoring the denominator as follows

#(3x)/(x-3)^2#

We have a repeated linear factor in the denominator, so our decomposition will take the following form:

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

Multiplying both sides by #(x-3)^2#

#3x=A(x-3)+B#

Distributing the #A#

#3x=Ax-3A+B#

Now equate

#Ax=3x# so #A=3#

#-3A+B=0#

#-3(3)+B=0#

#-9+B=0# so #B=9#

Now write in #A# and #B# into our decomposition

#(3x)/(x-3)^2=3/(x-3)+9/(x-3)^2#