grandes-ecoles 2021 Q8

grandes-ecoles · France · centrale-maths2__mp Factor & Remainder Theorem Remainder Theorem with Composed or Shifted Arguments

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda$ be the quotient of $P_\lambda = P(\lambda X) - P(\lambda)$ by $X-1$: $$Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$$ Show that, for all $\lambda$ in $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda$ be the quotient of $P_\lambda = P(\lambda X) - P(\lambda)$ by $X-1$:
$$Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$$
Show that, for all $\lambda$ in $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.

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