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))
