In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
In this case, it is actually possible to directly solve these two equations and then it will become obvious which two points have the largest distance.
For the first equation: $$z^3 -8=0\implies z=2, 2(\cos 120^{\circ} \pm i\cos 120^{\circ})$$
For the second equation: $$z^3 - 8z^2 - 8z + 64 = 0 \implies (z-8)(z^2-8)=0\implies z= 8, 2\pm\sqrt{2}$$
Plotting these points on the plane, it will be clear the biggest distance is between $2(\cos 120^{\circ}\pm i\sin 120^{\circ})$ and $8$. This distance can be computed using law of cosines: $$\sqrt{2^2 + 8^2 - 2\times 2\times 8\cos 120^{\circ}}=\boxed{2\sqrt{21}}$$