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Let \(x_1, \dots, x_n \in \mathbb{K}\) be pairwise distinct and \(L_1, \dots, L_n\) the associated Lagrange polynomials. Show that for every \(P \in \mathbb{K}_{n-1}[X]\), $$ P = \sum_{i=1}^{n} P(x_i) \, L_i. $$ Hint. Both sides are polynomials of degree at most \(n - 1\) that agree at the \(n\) distinct points \(x_1, \dots, x_n\).
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