Exercise 4, § 3

Algebra, IV, p. 89

Show that the Lagrange interpolation formula (IV, p. 16, Prop. 6) may be written

f(X) = \omega(X) \sum_{i=1}^n \frac{\beta_i}{\omega'(\alpha_i)(X-\alpha_i)}

where $\omega(X) = (X-\alpha_1)(X-\alpha_2) ... (X-\alpha_n)$ . Deduce that if g is a polynomial of K[X] of degree \leq n-2, then

\sum_{i=1}^n \frac{g(\alpha_i)}{\omega'(\alpha_i)} = 0

(consider the polynomial f(X) = Xg(X)).

Solution

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