Hey all, I need to negate the following statements, can I get feedback on if this is correct?

Every cubic polynomial has a real root.

Negation: There exists a cubic polynomial that has no real root.

G is normal and H is regular.

Negation: G is not normal or H is not regular.

There does not exist 0 such that for all x, x+0=x

Negation: For no 0 does there exist x, where x+0 =/ x

The newspaper article was neither accurate nor entertaining.

Negation: The newspaper article was accurate and entertaining.

If gcd(m,n) is odd, then m or n is odd.

Negation: Gcd(m,n) is odd and m is not odd and n is not odd.

H/N is a normal subgroup of G/N if and only if H is a normal subgroup of G.

Negation: H/N is a normal subgroup of G/N and H is not a normal subgroup of G or H is a normal subgroup of G and H/N is not a normal subgroup of G/N.

For each epsilon > 0 there exists N in the Natural Numbers such that for all n >= N, the abs|An - L| < epsilon

Negation: There exists epsilon > 0 for each N in the Natural Numbers such that there exists n >= N, |An - L| >= epsilon

Any help would be appreciated!