Let$ \langle p(x)\rangle $ denote the ideal generated by the polynomial $p(x)$ in $\mathbb Q[x]$. If $f (x) = x^3 + x^2 + x +1$ and $g(x) = x^3 – x^2 + x -1$, then which of the followings are true?
1. $ \langle f (x)\rangle + \langle g (x)\rangle = \langle x^3 + x\rangle$
2. $ \langle f (x)\rangle + \langle g (x)\rangle = \langle f (x)\cdot g (x)\rangle$
3. $ \langle f (x)\rangle + \langle g (x)\rangle = x^2 +1$
4. $ \langle f (x)\rangle + \langle g (x)\rangle = \langle x^2 -1\rangle$
Here gcd of them is $x^2+1$ so 3 is true and 4 is false. but I am not sure about the others. Can anybody help me.