On the parity of the coefficients from polynomials whose zeros are integral

Marcelo POLEZZI [1]

Trajano Pires da NÓBREGA NETO [2]

§     ABSTRACT: Consider a polynomial f(x)=(x-x1) ... (x-xn), where x1, … , xn Î Z. In this paper, we shall show three results which relate the number of even/odd coefficients of f(x) with the number of even/odd zeros of f(x). These results are closely connected to Girard's relations (due to the French-Flemish mathematician Albert Girard, 1595-1632) and to a divisibility criterion for binomial coefficients.

§     KEYWORDS: Polynomials whose zeros and coefficients are integral; Girard's relations; divisibility criterion for binomial coefficients.


[1] Universidade Estadual de Mato Grosso do Sul - UEMS, CEP 79540-000, Cassilândia, MS, Brasil. E-mail: mpolezzi@terra.com.br

[2] Departamento de Matemática, Universidade Estadual Paulista - UNESP, CEP 15054-000, São José do Rio Preto, SP, Brasil. E-mail: trajano@ibilce.unesp.br