Find example of nonhomogeneous ideal in $\mathbb{k}[x]$ which has two different minimal generating sets are not equipotent.

Question

I have problem with finding example of nonhomogeneous ideal in $\mathbb{k}[x]$ with two different minimal generating sets which are not equipotent.

Is there any general way to find such examples?

Is for example ideal generated by $x^7+1$ and $x^7-1$ a good example as it is also generated by $x^7$ alone?