Let $a_{1}, \ldots a_{n}$ and $b_{1}, \ldots, b_{n}$ be elements of $\mathbb{K}$ (with the $a_{i}$ pairwise distinct). Find all polynomials $P \in \mathbb{K}[X]$ such that $P\left(a_{i}\right)=b_{i}$.
We have already solved the problem when the degree of $\mathrm{P}$ is at most $\mathrm{n}-1$ thanks to the Lagrange interpolators. If the degree of $P$ is greater than or equal to $n$, let $L$ be the interpolating polynomial associated with the $a_{i}$ and $b_{i}$. The polynomial $P-L$ vanishes at $a_{1}, \ldots, a_{n}$. Therefore, we have

$$
P(X)=c_{1} \sum_{i=1}^{n} b_{i} \prod_{j=1, j \neq i}^{n} \frac{X-a_{j}}{a_{i}-a_{j}}+c_{2}\left(X-a_{1}\right)\left(X-a_{2}\right) \cdots\left(X-a_{n}\right)
$$