
Notons $P(x)=ax^3+bx^2+cx+d$ le polynôme cherché, et $\mathcal{C}$ sa courbe représentative.

\begin{itemize}
\item $\begin{aligned}[t]
A\coord{-2}{-1}\in\mathcal{C}& \iff P(-2)=-1\\
& \iff a(-2)^3+b(-2)^2+c(-2)+d=-1\\
& \iff -8a+4b-2c+d=-1.
\end{aligned}$

\item $\begin{aligned}[t]
B\coord{-1}{3}\in\mathcal{C}& \iff P(-1)=3\\
& \iff a(-1)^3+b(-1)^2+c(-1)+d=3\\
& \iff -a+b-c+d=3.
\end{aligned}$

\item $\begin{aligned}[t]
C\coord{1}{5}\in\mathcal{C}& \iff P(1)=5\\
& \iff a(1)^3+b(1)^2+c(1)+d=5\\
& \iff a+b+c+d=5.
\end{aligned}$

\item $\begin{aligned}[t]
D\coord{2}{34}\in\mathcal{C}& \iff P(2)=34\\
& \iff a(2)^3+b(2)^2+c(2)+d=34\\
& \iff 8a+4b+2c+d=34.
\end{aligned}$
\end{itemize}
En notant:
\[
A = \begin{pmatrix}
-8 & 4 & -2 & 1\\
-1 & 1 & -1 & 1\\
1 & 1 & 1 & 1 \\
8 & 4 & 2 & 1
\end{pmatrix}\quad,\quad X=\begin{pmatrix}a\\b\\c\\d\end{pmatrix}\quad,\quad B=\begin{pmatrix}-1\\3\\5\\34\end{pmatrix}
\]
On a :
\[
AX = B \iff X = A^{-1}B.
\]
À l'aide de la calculatrice, on obtient:
\[
\boxed{P(x) = \frac{31}{12}x^3+\frac{25}{6}x^2-\frac{19}{12}x-\frac{1}{6}}
\]
