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

##### 1 Answer

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

The next step is to multiply both sides of the equation by

#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^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

This implies that

Immediately from the last two equations, we can conclude that

From this it follows that since

Then, after plugging

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.