Further, since , at least two of the sets  and C are non-empty and by computing the above GCDs we may obtain non-trivial factors. Since the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm.
Given that at least two sets A and C are non-empty, calculating their greatest common divisors (GCD) yields non-trivial factors. This process is possible because the ring of polynomials over a field is a Euclidean domain, allowing for efficient computation using the Euclidean algorithm.