Inverse Laplace Transform of Third Order Denominator $\frac{s^2 -s -15} { s^3 +8s^2 +24s -32}$

Question

I am trying to find the inverse laplace transform of the following function: $$ f(s)=\frac{s^2 -s -15} { s^3 +8s^2 +24s -32} $$ I can't seem to factor the denominator to use partial fractions. Is there something I'm missing or another method to solve this? This is for a circuits class.

Answer 1

I cannot seem to find a nice expression for the roots, so I will outline a general implicit approach. Assuming that $\zeta_1,\zeta_2,\zeta_3$ are three distinct complex numbers and $q(s)$ is a second degree polynomial, non-vanishing over $\{\zeta_1,\zeta_2,\zeta_3\}$, $$ f(s)=\frac{q(s)}{(s-\zeta_1)(s-\zeta_2)(s-\zeta_3)}=\sum_{k=1}^{3}\frac{\eta_k}{s-\zeta_k}\tag{1} $$ where $$ \eta_k = \text{Res}_{s=\zeta_k}f(s) = \lim_{s\to \zeta_k}\frac{q(s)(s-\zeta_k)}{(s-\zeta_1)(s-\zeta_2)(s-\zeta_3)}=\frac{q(\zeta_k)}{(-1)^k \sqrt{D}}\tag{2} $$ with $D$ being a constant depending only on the discriminant of $(s-\zeta_1)(s-\zeta_2)(s-\zeta_3)$.
By applying $\mathcal{L}^{-1}$ to the RHS of $(1)$ and exploiting $(2)$ we get $$ (\mathcal{L}^{-1} f)(x) = \frac{1}{\sqrt{D}}\sum_{k=1}^{3}(-1)^k q(\zeta_k) e^{\zeta_k x}.\tag{3} $$ In particular, the behaviour of $\mathcal{L}^{-1}f$ in a left neighbourhood of $+\infty$ and the positions of $\zeta_1,\zeta_2,\zeta_3$ are deeply connected. In the given case $\zeta_1$ is close to $1$ and $\zeta_{2,3}$ are close to $-4.5\pm 3.5i$, hence $(\mathcal{L}^{-1} f)(x)$ is pretty close to $-\frac{15}{43}e^x$ for large values of $x$.