How do I find the partial fraction decomposition of #(x^4+1)/(x^5+4x^3)# ?

1 Answer
Aug 13, 2014

First we will factor the denominator as much as possible:

#(x^4 + 1)/(x^3(x^2 + 4))#

And now, we will choose the factors to write:

#(x^4 + 1)/(x^3(x^2 + 4)) = A/x + B/x^2 + C/x^3 + (Dx+E)/(x^2 + 4)#

Note that since there was a lone power of #x#, (which was #x^3#) I wrote out successive powers of #x#, starting at #x# to the first, and ending at #x^3#. There was also a quadratic term, #x^2 + 4#, which couldn't be factored - so for that one, we used #Dx + E# in the numerator.

The next step is to multiply both sides of the equation by #x^3*(x^2 + 4)# and cancel off what we can:

#x^4 + 1 = Ax^2*(x^2 + 4) + Bx*(x^2+4) +#
#C*(x^2 + 4) + x^3(Dx+E)#

And now, we will distribute and simplify everything:

#x^4 + 1 = Ax^4 + 4Ax^2 + Bx^3 + 4Bx + Cx^2 +#
#4C + Dx^4 + Ex^3#

We can solve for each constant now, by using the technique of grouping. The first step is to rearrange everything in successive powers of #x#:

#x^4 + 1 = Ax^4 + Dx^4 + Bx^3 + Ex^3 + 4Ax^2 + Cx^2+#
#4Bx + 4C#

And now, we will factor out the constant terms:

#x^4 + 1 =(A + D)x^4 + (B+E)x^3 + (4A+C)x^2 + 4Bx + 4C#

The next step is to create a system of equations using the coefficients of #x# on the left side that correspond to the coefficients of #x# on the right side. What do I mean? Well, we can see that there is a term #1*x^4# on the left side, but there is also a #(A + D)*x^4# term on the right side.

This implies that #A + D = 1#. We will continue in this manner, building a system using all the coefficients:

#A + D = 1#
#B+E = 0#
#4A+C = 0#
#4B = 0#
#4C = 1#

Immediately from the last two equations, we can conclude that #B = 0# and #C = 1/4#.

From this it follows that since #B + E = 0#, #E# must also equal #0#. And since #4A + C = 0#, #A# must equal #-1/16#.

Then, after plugging #A# into the last equation #A + D = 1#, and solving for #D#, we obtain #D = 17/16#.

Now all that's left is to plug these coefficient values into our expanded expression:

#(x^4 + 1)/(x^3(x^2 + 4)) = A/x + B/x^2 + C/x^3 + (Dx+E)/(x^2 + 4)#

#(x^4 + 1)/(x^3(x^2 + 4)) = 1/(4x^3) + (17x)/(16(x^2 + 4)) - 1/(16x)#

And there we have it. Remember, successfully expanding with partial fractions is all about choosing the correct factors, and from there it's just a lot of algebra. If you are familiar with the grouping technique, then you shouldn't have any trouble solving for the coefficients.