The title says it all: I want to show that an arbitrary cubic polynomial can be factored as a product of linear terms without appealing to the fundamental theorem of algebra and (preferably) without appealing to the general formula for solving a cubic equation. This is exercise $2.11$ from Ian Stewart's Galois Theory text. Explicitly, it says
Prove, without using the winding number, that every cubic polynomial over $\Bbb C$ can be expressed as a product of linear factors over $\Bbb C$.
In the exercise before, he asks us to show that a cubic polynomial over $\Bbb R$ can be factored as a product of linear terms over $\Bbb C$. This is pretty much immediate since it necessarily tends to $-\infty$ in one direction and $\infty$ in the other direction, hence by the IVT it has a zero. The remainder theorem says that we can factor this linear term out, and then we just use the quadratic formula and we're done. I've been thinking about how we can show something similar for a polynomial over $\Bbb C$ and I haven't figured anything out. Is there a method besides appealing to the formula for cubic polynomials?
The proof of the FTA in this book uses the winding number and presents the general formula for a cubic, so I'm not inclined the solution for this problem relies on either of these results.