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If a is selected as a = xj for some j i, then x!) contains the value xj at two distinct
index locations in its list (i.e. at i and j). Consequently, if y(x) represents the value of the
Vandermonde determinant in Eq. C.2 (i.e. (x) = A,(x) = V.(x) ), then the Vandermonde
determinant represented by the polynomial function Vy(-) = An,,a) = I V,(ni)) contains
two identical rows and hence is zero. In that case, the Bezout theorem reduces to
(x) = (xi x)().
Thus (x, xj) is a factor of An(x) = I V,(x) .
This argument applies to each of the n(n 1)/2 distinct difference factors (x, x). It
follows that A,(x) = I V,(x)[ can be written as
p=n
An(x) = I V,(x)l = (x) n (xP xq) Eq. C.3
p=2,q