Polynomial Rings: Finding 8 Elements with r^2=r

[jgens]

Homework Statement

Find eight elements $r \in \mathbb{Q}[x]/(x^4-16)$ such that $r^2=r$.

Homework Equations

N/A

The Attempt at a Solution

The elements $0+(x^4-16)$ and $1+(x^4-16)$ clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a technique for finding a few more elements?

Thanks!

 

[micromass]

You can always brute force it. An arbitrary element in your ring has the form $[ax^3+bx^2+cx+d]$. Square it and see when the relation is satisfied.