We consider the application $\Delta$ defined by $\Delta(P) = P(X+1) - P(X)$. Deduce $\operatorname{Ker}(\Delta)$ and $\operatorname{Im}(\Delta)$. Apply the results obtained to the study of the equation $(E_{h})$:
$$\forall x \in \mathbb{K},\, f(x+1) - f(x) = h(x)$$
in the case where $h$ is a polynomial function.