Try to deal with a problem, that might actually be related to Laplace transform. Here is brief overview. Let $P(t)={p}_{m}{t}^{m}+\cdots +{p}_{1}t+{p}_{0}$. Then we know that,it is possible to express the following integral in the form.

$${\int}_{0}^{\mathrm{\infty}}{e}^{-zt}P(t)=\sum _{a=0}^{m}{p}_{a}\frac{a!}{{z}^{a+1}},$$

where the equality comes from the property of Laplace transform.

$${\int}_{0}^{\mathrm{\infty}}{e}^{-zt}P(t)=\sum _{a=0}^{m}{p}_{a}\frac{a!}{{z}^{a+1}},$$

where the equality comes from the property of Laplace transform.