Chapter 21 Deducing Polynomial Division Algorithms Using a Groebner Basis

A computerized algebra algorithm is used to find polynomials and the integer coefficients a, b and n for which the polynomial F(x) = a x2n + bxn + Dn – (Bx2+Cx+D)n is divisible by G(x) = x3 +Bx2+Cx+D.  Consequently, x is any of the real or complex roots of G such that G(x) = F(x) = 0.  Also, the coefficients a, b are integers derived from the integer sequence associated with the cubic polynomial G. Higher order polynomials divisible by G(x) are also derived using a Groebner basis. It is shown that new recurrence relationships can be generated from the Groebner basis.

Chapter 21— Deducing Polynomial Division Algorithms

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