It is shown that a real Hankel matrix admits an approximate block diagonalization inwhich the successive transformation matrices are upper triangular Toeplitz matrices. Thestructure of this factorization was first fully discussed in [1]. This approach isextended to obtain the quotients and the remainders appearing in the Euclidean algorithmapplied to two polynomials u(x) andv(x) of degree n andm, respectively, whith m <n