My teacher gave the following hint:

The theorem states the existence of such two functions, and we don't know what they are. What we know is that there ARE two such functions. That's what the theorem is about, namely, the existence of such two functions. Now, not knowing what those are, we pull out tricks and managed to find two solutions; for example, z1=t² and z2=t³. Using this example, I raise a question: are we going to reach all the solutions by considering

a t² + b t³

and all constants a and b?