Is this proof right for The Fundamental Theorem of Algebra with matrix theory?

Question

$\forall f(x) = \sum _{i = 0} ^{n} a_i x^i, (n \ge 1, a_i \in \mathbb C)$ Let $g(x) = \dfrac {1} {a_n} f(x),$ then:
$g(\lambda) = \left \vert \lambda I - A \right \vert $ ($A$ is a Frobenius companion matrix)
$= \left \vert \lambda I - P^{-1}JP \right \vert $ ( $J$ is a Jordan matrix)
$= \vert P^{-1} \vert \left \vert \lambda I - J \right \vert \vert P \vert $ $= \left \vert \lambda I - J \right \vert $ $= \prod _{i = 1} ^{n} (\lambda - \lambda_i )$
$\Rightarrow f(\lambda) = a_n g(\lambda) = a_n \prod _{i = 1} ^{n} (\lambda - \lambda_i )$

Answer 1

How do you prove the existence of the Jordan normal form without assuming the Fundamental Theorem of Algebra?