1. Suppose that $P$ is a unitary polynomial of degree $\leq 4$ satisfying $P(1)=1, P(2)=4, P(3)=9, P(4)=16$. Find $P$.
2. Suppose that $P$ is a polynomial such that $P\left(X^{3}\right)=P(X)^{3}$ and $P(2)=2$. Find $P$.
1. We can factorize: $P-X^{2}=C \cdot(X-1)(X-2)(X-3)(X-4)$ with $C$ a constant.

If $C \neq 0$, since $P$ is of degree 4 and monic, $C=1$.

Hence $P=X^{2}$ or $P=X^{2}+(X-1)(X-2)(X-3)(X-4)$.

2. We prove by induction that $P\left(2^{3^{n}}\right)=2^{3^{n}}$ so $P-X$ has an infinite number of roots, hence $P(X)=X$.

## Derivative Polynomial and Analytical Properties

The derivative polynomial can essentially be calculated in two ways:

- If $P(X)=\sum_{i=0}^{n} a_{i} X^{i}$, then $P^{\prime}(X)=\sum_{i=1}^{n} i a_{i} X^{i-1}$
- If $P(X)=C \cdot \prod_{i=1}^{r}\left(X-\alpha_{i}\right)^{m_{i}}$, then $P^{\prime}(X)=\sum_{i=1}^{r} m_{i}\left(X-\alpha_{i}\right)^{m_{i}-1} \prod_{j \neq i}\left(X-\alpha_{j}\right)^{m_{i}}=\sum_{i=1}^{r} \frac{m_{i} P(X)}{X-\alpha_{i}}$

The derivative polynomial is particularly interesting because in the context of real polynomials, it gives the slope of the tangent line to the curve representing the function $x \in \mathbb{R} \mapsto P(x)$. In particular, the sign of the derivative polynomial indicates whether the function is strictly increasing $(>0)$, or strictly decreasing $(<0)$ on an interval. When the polynomial reaches a maximum or minimum on an interval, its derivative polynomial vanishes. The derivative polynomial thus gives the variations of $P$.

For a polynomial $P$, we denote $P^{\prime}$ its derivative polynomial.

We can then define the successive derivatives of $P$ by: $P^{(0)}=P$ and for all $m \geq 0, P^{(m+1)}=\left(P^{(m)}\right)^{\prime}$.

Derivation has some properties: if $P$ and $Q$ are polynomials:

- $(P+Q)^{\prime}=P^{\prime}+Q^{\prime}$
- $(P \cdot Q)^{\prime}=P^{\prime} Q+P Q^{\prime}$
- $P^{\prime}=0 \Longleftrightarrow P$ is constant.

We can characterize the multiplicity of a root $\alpha$ of $P$ with the successive derivatives of $P$: if $m \in \mathbb{N}^{*}, \alpha$ is a root of multiplicity $m$ of $P$ if, and only if, $\alpha$ is a root of $P, P^{\prime}, \ldots, P^{(m-1)}$ but not of $P^{(m)}$.