$ f$ and $g$ generate an ideal I $\subset \mathbb{Z}[x]$ show that we can replace the generators with two new generators such that one of them has zero constant term.
I know $f,g-hf$ also generate the ideal where h is any element of $\mathbb{Z}[x]$. I also know if it exists the remaining constant term must be the gcd of the original constant terms and I know the gcd can be written as a linear combination of these original terms.
Lets call the constant terms F and G. Then zF+wG=gcd(F,G). Maybe I can use this to write my new generators somehow?