\(x_0\) est racine d’ordre de multiplicité 3 ssi \(P\left(x_0\right)=P^{\prime}\left(x_0\right)=P^{\prime \prime}\left(x_0\right)=0\) et \(P^{(3)}\left(x_0\right) \neq 0\). \[\begin{aligned} & P\left(x_0\right)=3 \cdot x_0^5-10 \cdot x_0^3+a \cdot x_0+b \\ & P^{\prime}\left(x_0\right)=15 \cdot x_0^4-30 \cdot x_0^2+a=0 \\ & P^{\prime \prime}\left(x_0\right)=60 \cdot x_0^3-60 x_0=60 \cdot x_0\left(x_0^2-1\right)=0 \\ & P^{(3)}\left(x_0\right)=180 \cdot x_0^2-60=60\left(3 \cdot x_0^2-1\right) \end{aligned}\] (3) \(\Leftrightarrow x_0 \cdot\left(x_0^2-1\right)=0 \Leftrightarrow x_0=0\) ou \(x_0=1\) ou \(x_0=-1\).

\(1^{\text {er }}\) cas: \(x_0=0\) \[\begin{gathered} P^{(3)}\left(x_0\right) \neq 0 \\ (2) \Rightarrow a=0 \\ (1) \Rightarrow b=0 \\ P(X)=3 \cdot X^5-10 \cdot X^3=X^3 \cdot\left(3 X^2-10\right)=3 \cdot X^3 \cdot\left(X-\sqrt{\frac{10}{3}}\right)\left(X+\sqrt{\frac{10}{3}}\right) \end{gathered}\] \(2^{\text {ème }}\) cas: \(x_0=1\) \[\begin{gathered} P^{(3)}\left(x_0\right) \neq 0 \\ (2) \Rightarrow 15-30+a=0 \Rightarrow a=15 \\ (1) \Rightarrow 3-10+15+b=0 \Rightarrow b=-8 \\ P(X)=3 \cdot X^5-10 \cdot X^3+15 \cdot X-8=(X-1)^3 \cdot\left(3 X^2+9 X+8\right) \end{gathered}\] \({3}^{\text {ème }}\) cas: \(x_0=-1\) \[\begin{gathered} P^{(3)}\left(x_0\right) \neq 0 \\ (2) \Rightarrow 15-30+a=0 \Rightarrow a=15 \\ (1) \Rightarrow-3+10-15+b=0 \Rightarrow b=8 \\ P(X)=3 \cdot X^5-10 \cdot X^3+15 \cdot X+8=(X+1)^3 \cdot\left(3 X^2-9 X+8\right) \end{gathered}\]

Multiplicité et factorisation