Proof that complex polynomials of degree three are equivalent under affine transformations to $z^3$ or $z^3-3z$

Question

I'm having a hard time showing that any complex polynomial of degree three can be turned into either $z^3$ or $z^3 - 3z$ with an affine change of coordinates in the domain and the codomain.
The definition given for an affine transformation is one that looks like $w=\alpha z + \beta$ where $\alpha\neq0$.
In the real case this problem doesn't seem very complicated but I'm just starting with complex analysis and I don't see how to approach the problem clearly. And I couldn't find another question which could help me. Thanks in advance.

Answer 1

The polynomial

$$az^3+bz^2+cz+d$$ can be turned to

$$z^3+pz+q$$ by reducing the coefficients and with a shift of the argument.

By subtracting the constant,

$$z^3+pz.$$

If $p=0$, we have $z^3$.

Otherwise, with the scaling $z\to sz$ such that $p=-3s^2$,

$$s^3z^3+psz=s^3(z^2-3z).$$

All the above transformations are affine, on $z$ or on $p(z)$.