Can You Always Factor the Denominator in Partial Fraction Expansion?

[Runei]

Now this is a pretty straight forward question. And I just want to make sure that I am not doing anything stupid.

But when doing partial fraction expansions of the type

$\frac{K}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$ Shouldnt I always be able to factor the denominator into the following:

$\left(s-s_{1}\right)\left(s-s_{2}\right)$

where

$s_{1} = -\zeta\omega_{n}+\omega_{n}\sqrt{\zeta^{2}-1}$ and
$s_{2} = -\zeta\omega_{n}-\omega_{n}\sqrt{\zeta^{2}-1}$

And thus being able to make the following expansion:

$\frac{A}{s-s_{1}}+\frac{A}{s-s_{2}} = \frac{K}{\left(s-s_{1}\right)\left(s-s_{2}\right)}$

Since s1 and s2 are the roots of the polynomial?

These roots may ofcourse either be real and distinct, repeated or complex conjugates.

Thank in advance,
Rune

 

[Runei]

The reason for the question is that I am reading for an exam for Control Systems. And I am using LaPlace transforms to solve the differential equations.

To get back to the time-domain I am using partial fraction expansions, and for example right now I am trying to do the partial fraction expansion of

$\frac{K}{s\cdot\left(s^{2}+2\zeta\omega_{n}s + ω_{n}^{2} \right)}$

And I am trying to determine whether I am actually doing it wrong when factoring, or whether I can actually solve the problem by expanding to the following:

$\frac{A}{s}+\frac{B}{s-s_{1}}+\frac{C}{s-s_{2}} = \frac{K}{s\cdot\left(s-s_{1}\right)\cdot\left(s-s_{2}\right)}$

 

[Mark44]

This factorization is fine, as long as there are no repeated real roots in the quadratic.

Consider the case where the denominator is s(s² + 4s + 4). Here is the partial fraction decomposition:

$$ \frac{K}{s(s^2 + 4s + 4)} = \frac{A}{s} + \frac{B}{s + 2} + \frac{C}{(s + 2)^2}$$

 

[Runei]

Thank you Mark!

I actually found an error further back in my work and that was because I didn't get the correct result. But thank you for clarifying and making me sure :)