How many zeroes do the polynomials
P1(x)=x3+x2+x+1 and
P2(x)=x2+1
have in common?
0
P1(x)=x3+x2+x+1
=(x+1)(x2+1).
So, There are two factors of P1(x)
On the other hand, P2(x) cannot be factorised.
Thus, P1(x) and P2(x) have no zeroes in common.