The great trinomial hunt

A trinomial is a polynomial in one variable with three nonzero terms, for example P = 6x + 3x − 5. If the coefficients of a polynomial P (in this case 6, 3,−5) are in some ring or field F , we say that P is a polynomial over F , and write P ∈ F [x]. The operations of addition and multiplication of polynomials in F [x] are defined in the usual way, with the operations on coefficients performed in F . Classically the most common cases are F = Z,Q,R or C, respectively the integers, rationals, reals or complex numbers. However, polynomials over finite fields are also important in applications. We restrict our attention to polynomials over the simplest finite field: the field GF(2) of two elements, usually written as 0 and 1. The field operations of addition and multiplication are defined as for integers modulo 2, so 0 + 1 = 1, 1 + 1 = 0, 0× 1 = 0, 1× 1 = 1, etc. An important consequence of the definitions is that, for polynomials P,Q ∈ GF (2)[x], we have