I have a system matrix (A): $$ A =\begin{bmatrix} 0 & 1 & 0 \\ -C/I & 0 & B/I \\ 0 & -D/K & -E/K \\ \end{bmatrix} $$
I'm taking the determinant: $$ \begin{vmatrix} sI_{n}-A \end{vmatrix} = 0 $$ where $I_{n}$ is the identity matrix. This yields:
$$ \begin{vmatrix} s & -1 & 0 \\ C/I & s & -B/I \\ 0 & D/K & (s+E/K) \\ \end{vmatrix} = 0 $$
This leads to: $$ s^2(s+\dfrac{E}{K}) + s(\dfrac{BD}{IK}) + \dfrac{C}{I}(s+\dfrac{E}{K}) $$
Expanding: $$ s^3 + (\dfrac{E}{K})s^2 + (\dfrac{C}{I}+\dfrac{BD}{KI})s+\dfrac{E}{K} $$
Leaving me with a 3rd order polynomial in $s$.
Solving this now for $s$ is what I would like to do next, in order to determine the poles - i.e., solutions of $ s^3 + (\dfrac{E}{K})s^2 + (\dfrac{C}{I}+\dfrac{BD}{KI})s+\dfrac{E}{K} = 0 $. If anyone can help with this it would be very much appreciated.