\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Going further. Let \(P \in \mathbb{R}[X]\) be a polynomial of degree \(n\) that splits with simple real roots \(\lambda_1, \ldots, \lambda_n\). Show that for every \(k \geq 0\) and every \(x \in \mathbb{R}\) avoiding the \(\lambda_i\): $$ \left( \frac{1}{P} \right)^{(k)} (x) = (-1)^k k! \sum_{i = 1}^{n} \frac{1}{P'(\lambda_i) (x - \lambda_i)^{k + 1}}. $$
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