**Partial fraction decomposition** is the reverse of the process normally used to add fractional expressions with different denominators.

We want to find values #A# and #B# such that

#color(white)("XXX")A/(x+3)+B/(x-5) = (x+11)/((x+3)(x-5))#

That is

#color(white)("XXX")(A(x-5)+B(x+3))/((x+3)(x-5)) = (x+11)/((x+3)(x-5))#

#color(white)("XXX")A(x-5)+B(x+3)=x+11#

#color(white)("XXX")Ax+Bx=xcolor(white)("XXX")rArr#

[1]#color(white)("XXX")A+B = 1#

and

[2]#color(white)("XXX")-5A+3B=11#

Rewriting [1] as

[3]#color(white)("XXX")B=1-A#

Substitute #(1-A)# for #B# in [2]

[4]#color(white)("XXX")-5A+3(1-A)=11#

[5]#color(white)("XXX")-8A+3=11#

[6]#color(white)("XXX")-8A=8#

[7]#color(white)("XXX")A=-1#

Substituting #(-1)# for #A# in [1]

[8]#color(white)("XXX")B=2#

Referring back to our original equation

#color(white)("XXX")A/(x+3)+B/(x-5)= (-1)/(x+3)+(2)/(x-5)#

#color(white)("XXXXXXXXXXXX") = (x+11)/((x+3)(x-5))#