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))#
