Not exactly sure what to do here with so many possibliities. I've found random one off cases, but no patterns emerge. Help Please!

This is easy. Try starting with linear polynomials.

How would I prove that they are the only ones that exist? the theme of the week is being able to prove rigously and also to be able to prove it to an algebra 1 student.

I'd like to see this one solved too (with plenty of explanation).

Anyone interested?