Prove cubic equation on $\mathbb{C}$ can be expressed as a product of linear factors

Question

I want to ask about Exercise 2.8 in Ian Stewart's Galois Theory states that: "Without using Fundamental Theorem of Algebra, prove that a cubic polynomial on $\mathbb{C}$ can be expressed as a product of linear factors".

Exercise 2.7 states a similar question but we consider the cubic polynomial on $\mathbb{R}$. I solved it by noting that it must have a root in $\mathbb{R}$, and any quadratic polynomial with coefficients in $\mathbb{R}$ can be expressed as a product of linear factors.

Obviously, the same idea can't be applied in Exercise 2.8. Help me with this

Answer 1

I'm guessing you can still use the fact that cube roots and square roots exist in $\mathbb{C}$. Then given a cubic, you can apply Cardano's method (or just a cubic formula https://en.wikipedia.org/wiki/Cubic_function). This will give you one root, say $z_1$. Then you can factor your polynomial as follows: \begin{equation*} f(x)=(x-z_1)g(x) \end{equation*} where $g(x)$ is a quadratic. Then apply the quadratic formula to $g$ to get the remaining factors.