How to show that $(3,x-1)\not=(3,x+1)$ as ideals in $Z[x]$

Question

How to show that $(3,x-1)\not=(3,x+1)$ as ideals in $Z[x]$

Both are maximal.

I think all i need to do is to show is that $x+1 \not \in (3,x-1)$ but I do not know how to show that. I wanted to assume towards contradiction that $3f(x)+(x-1)g(x)=x+1$. It is clear that $deg(f(x))\geq1$ and $deg(g(x))=deg(f(x))-1$. I was thinking induction on the degree but it seems messy with weird system of equations.

Answer 1

Hint: If $3f(x)+(x-1)g(x)=x+1$, then what happens when $x=1$?