Question

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

Answer 1

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.