A proof of the main theorem on Bezoutians BrankoĆurgus and Aad Dijksma B.Ćurgus has a PhD in mathematics from the University of Sarajevo. He enjoys teaching and researching beautiful mathematics. A. Dijksma (PhD 1971) taught mathematics at the University of Groningen from 1972 to 2005. His research concerns operator theory including spectral theory and Schur analysis. With two polynomials f and g and n = max{deg f, deg g} we associate an n ×n matrix B, called the Bezoutian, and a 2n × 2n matrix

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... R, called the resultant. Their defining relations are given by (6) and (5) below, respectively. In terms of the coefficients of f and g they are given by (8) and (4) . In this note we give a simple and self-contained proof of the where gcd stands for greatest common divisor. H.K. Wimmer in [8] attributes this result to Jacobi who in 1836 showed that the singularity of what we call the Bezoutian implies the existence of a common factor of f and g. More contemporary proofs of (1) can be found in the recent books [3, Theorems 21.10 and 21.11] by H. Dym and [5, Theorem 8.30] by P.A. Fuhrmann. In the Introduction to [3, Chapter 21] it is shown that . The FrenchmanÉtienne Bézout (1730-1783) taught mathematics at the Garde du Pavillon, the Garde de la Marine and the Corps d'Artillerie and wrote several textbooks used widely in Europe and the USA. The little time left for research he devoted mainly to solving systems of equations in several variables. He developed the "method of simplifying assumptions": when the general problem appears unsoluble consider first special problems by making assumptions. He was successful: a theorem in algebraic geometry, an identity in elementary number theory, an integral domain and a matrix now carry his name. Our note concerns the Bézoutian matrix so termed by the Copley medalists James Joseph Sylvester (in 1853) and Arthur Cayley (in 1857). The Bézoutian matrix is a square matrix associated with two polynomials whose nullity equals the number of their common zeros counting multiplicities. We give a selfcontained new proof of this fact. In the English literature the accent aigu on the e is often omitted.