The relationship between Bézoutian matrix and Newton’s matrix of divided differences

<p> Let x 1 ,...,x n be real numbers, P(x)=p n (x-x 1 )&ctdot;(x-x n ), and Q(x) be a polynomial of order less than or equal to n. Denote by &Delta;(Q) the matrix of generalized divided differences of Q(x) with nodes x 1 ,...,x n and by B(P,Q) the B&eacute;zoutian matrix of P and Q. A relationship between the corresponding principal minors of the matrices B(P,Q) and &Delta;(Q) counted from the right lower corner is established. It implies that if the principal minors of the matrix of divided differences of a function g(x) are positive or have alternating signs then the roots of the Newton&rsquo;s interpolation polynomial of g are real and separated by the nodes of interpolation.</p>