Evaluating the inverse laplace transform of $ X(s) = \frac{2\cdot a^4\cdot s}{s^4+4\cdot a^4} $.

Question

I'm preparing myself for the following exam, it's Signals and Systems. And I'm stuck with this example:

$$ X(s) = \frac{2\cdot a^4\cdot s}{s^4+4\cdot a^4} $$ The task is to find the inverse Laplace transform.

I know that I have to do Partial fraction decomposition, but I'm unable to find the roots for $s^4+4\cdot a^4$. The roots should be : $$s_1 = (-1-i)a $$ $$s_2 = (-1+i)a$$ $$s_3 = (1-a)a$$ $$s_4 = (1+a)a$$

Thanks in advance!

Answer 1

Hint:) \begin{align} s^4+4a^4 &=s^4+4a^4+4a^2s^2-4a^2s^2 \\ &=(s^2+2a^2)^2-4a^2s^2 \\ &=(s^2-2sa+2a^2)(s^2+2sa+2a^2)\\ &=((s-a)^2+a^2)((s+a)^2+a^2) \end{align}