How do you write the partial fraction decomposition of the rational expression #(7x-2)/((x-3)^2(x+1))#?

2 Answers
Jul 31, 2017

The answer is #=(-9/64)/(x-3)+(9/16)/(x-3)^2+(19/4)/(x-3)^3+(9/64)/(x+1)#

Explanation:

Let's perform the decomposition into partial fractions

#(7x-2)/((x-3)^2(x+1))=A/(x-3)+B/(x-3)^2+C/(x-3)^3+D/(x+1)#

#=(A(x-3)^2(x+1)+B(x-3)(x+1)+C(x+1)+D(x-3)^3)/((x-3)^2(x+1))#

The denominators are the same, we compare the numerators

#(7x-2)=A(x-3)^2(x+1)+B(x-3)(x+1)+C(x+1)+D(x-3)^3#

Let #x=3#, #=>#, #19=4C#, #=>#, #C=19/4#

Let #x=-1#, #=>#, #-9=-64D#, #=>#, #D=9/64#

Coefficients of #x^3#,

#0=A+D#, #=>#, #A=-D=-9/64#

Coefficients of #x^2#

#0=-5A+B-9D#

#=>#, #B=5A+9D=-45/64+81/64=36/64=9/16#

Therefore,

#(7x-2)/((x-3)^2(x+1))=(-9/64)/(x-3)+(9/16)/(x-3)^2+(19/4)/(x-3)^3+(9/64)/(x+1)#

Jul 31, 2017

#(7x-2)/((x-3)^2(x+1)) = color(blue)(9/(16(x-3)) + 19/(4(x-3)^2) - 9/(16(x+1))#

Explanation:

We recognize that there will be three different fractions with denominators of

  • #x-3#

  • #(x-3)^2#

  • #x+1#

So we set it up as

#(7x-2)/((x-3)^2(x+1)) = A/(x-3) + B/((x-3)^2) + C/(x+1)#

Multiply both sides by the quantity #(x-3)^2(x+1)#:

#7x-2 = A(x-3)(x+1) + B(x+1) + C(x-3)^2#

Expand the terms:

#7x-2 = A(x^2-2x+3) + B(x+1) + C(x^2-6x+9)#

Collect the terms by power of #x#:

#7x-2 = x^2(A+C) + x(-2A + B - 6C) + -3A + B + 9C#

Equate the coefficients on both sides (there is no #x^2# term on left hand side, so that we set equal to #0#:

#0 = A+C#

#7 = -2A + B - 6C#

#-2 = -3A + B + 9C#

Now we set up an augmented matrix (sorry for the poor quality):

#((1color(white)(aaa),0color(white)(aaa),1color(white)(aaa)|-2),(-2color(white)(aaa),1,-6color(white)()|7), (-3color(white)(aaa),1color(white)(aaa),9color(white)(aaa)|-2))#

In order to avoid mayhem, I won't work out the solving of the matrix (maybe you already know how, if not there's plenty of other sources, maybe some of Socratic!), but the solutions are

#A = 9/16#

#B = 19/4#

#C = -9/16#

Plugging these into the original equation, we have

#(7x-2)/((x-3)^2(x+1)) = color(blue)(ul(9/(16(x-3)) + 19/(4(x-3)^2) - 9/(16(x+1))#