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Prove the polynomial Taylor formula: for every \(P \in \mathbb{K}[X]\) of degree \(\leq n\) and every \(a \in \mathbb{K}\), $$ P(X) = \sum_{k=0}^{n} \frac{P^{(k)}(a)}{k!} (X - a)^k. $$ Hint. Set \(Q(Y) := P(Y + a) \in \mathbb{K}[Y]\), write \(Q(Y) = \sum_{k=0}^{n} c_k Y^k\), then identify \(c_k\) by computing \(Q^{(k)}(0)\).
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