Let $f(z)$ be a degree $3$ complex polynomial with real coefficients. How do we prove that $f(z)=0$ has three complex solutions without using the Fundamental Theorem of Algebra.
If $f(x)$ was real variable polynomial, I could use the Intermediate value theorem to at least get one root but with complex variable I have no idea what to do or how to start.
Best Answer
... tells you precisely that you can think of $f$ as a function $f\colon\mathbb R\to\mathbb R$ and can use intermediate value theorem.
After you obtain one real root $\alpha$, divide $f$ by $x-\alpha$ to get a degree $2$ polynomial. I assume you know how to prove how many roots it has.