A computerized algebra algorithm is used to find polynomials and the integer coefficients a, b and n for which the polynomial F(x) = a x^{2n} + bx^{n} + D^{n} – (Bx^{2}+Cx+D)^{n} is divisible by G(x) = x^{3} +Bx^{2}+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.