I=x-1 J=x+1 ideal of ring Z [x] ,I+J=Z[x]

Question

I=(x-1) J=(x+1) ideals of ring Z [x] , How can i demonstrate I+J=Z[x]? Probably I+J won't be equal to Z [x].

Answer 1

Hints:

You should think: which elements can we get inside an ideal, like $I$? Which elements can we get inside a sum of ideals, like $I+J$?

Then, $I+J=\mathbb Z[x]$ will be true if and only if $1\in I+J$, can you see why?

Do you know of some result which could help you in determining if $1$ is in $I+J$?