How do you write the partial fraction decomposition of the rational expression # (x^2 + 4)/ ((x^2 - 4)(x^2 +2))#?

1 Answer
Dec 14, 2015

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

Explanation:

Factor the denominator.

#(x^2+4)/((x+2)(x-2)(x^2+2))=A/(x+2)+B/(x-2)+(Cx+D)/(x^2+2)#

#x^2+4=A(x-2)(x^2+2)+B(x+2)(x^2+2)+(Cx+D)(x^2-4)#

If #x=-2#:

#8=-24A#
#A=-1/3#

If #x=2#:

#8=24B#
#B=1/3#

If #x=0#:

#4=-4A+4B-4D#
#1=-A+B-D#

Plug in #A# and #B# to see that #D=-1/3#.

If #x=1#:

#5=-3A+9B-3C-3D#

Plug in all the known values to find that #C=0#.

Plug these all back in to see that the expression decomposes into

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