From the fact that both c and d are a gcd, it follows that c | d and that d | c. Thus there is a polynomial e with d = ce. Sinc d | c, we have that deg(d) deg(c). This implies deg(e) = 0. Thus e is a constant, which is clearly nonzero.
deg(c).
This implies deg(e) = 0. Thus
e is a constant, which is clearly nonzero.