Let Fp denote the field Z/pZ for p a prime, and consider the ring Fp[x] of polynomials with coefficientsin Fp.

Let Fp denote the field Z/pZ for p a prime, and consider the ring Fp[x] of polynomials with coefficientsin Fp. We say that two polynomials g(x), h(x) are congruent modulo f(x) if f(x) divides g(x) − h(x)in Fp[x], i.e., there exists a polynomial q(x) in Fp[x] so that f(x)q(x) = g(x) − h(x). Now let p = 2 andlet f(x) = x2 + x + 1 and g(x) = x. Find a polynomial h(x) so that g(x)h(x) is congruent to 1 modulof(x). (You can do this by trial and error because this is a simple example, but you can also use theanalogue of the Euclidean algorithm for polynomials if you wish.)