I know that if $I$ is a homogeneous ideal in $A[x]$ then $\mathrm{ann}(I)$ is also homogeneous but I don't know how to prove it.
I have tried this:
Suppose $I$ is a homogeneous ideal and take $f \in \mathrm{ann}(I)$. Then for each $g \in I$ we have $fg=0$. If $f_1,...,f_n$ are the homogeneous components of $f$ and $g_1,...,g_m$ are the homogeneous components of $g$ then $fg_i=0$ for each $i=1,...,m$. I want to show that $f_ig=0$ for each $i=1,...,n$, but that's where I got stuck.