Laplace transform initial values with polynomial and impulse response

Question

Let $n ∈ N, n ≥ 1,$ and let $a_0...a_{n−1} ∈ R$ be fixed, $a_n = 1$. Let A be the linear differential operator given by

$A(y):=\sum\limits_{k=0}^n a_ky^k$

$(a_n=1,y^0=y)$

and p given by

$p(\lambda)=\sum\limits_{k:=0}^n a_k\lambda^k$

the characteristic is polynomial

The inverse Laplace transform of the funtion $C_0$ given (for s sufficiently large) by

$C_0(s):= \dfrac{1}{p(s)}$

This is the impulse response of the system given by A.

Show that $A[c_0] = 0$ and find the initial values

$c_0(0), c'_ 0(0), . . ... c^{n−1}(0).$

I understand that I have to find the ℒ$[A[c_0]](s)=0$ however I have no idea how to find this.

Could someone explain this question to me?