ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5631 on D4312: Complex polynomial function
Subresult of R4112: Strong fundamental theorem of complex algebra

Strong fundamental theorem of complex algebra for a quadratic complex polynomial

Formulation 0

Let $f : \mathbb{C} \to \mathbb{C}$ be a D4312: Complex polynomial function such that

(i)	$a, b, c \in \mathbb{C}$ are each a D1207: Complex number
(ii)	\begin{equation} f(z) = a z^2 + (a b + a c) z + a b c \end{equation}

Then \begin{equation} f(z) = a (z + b) (z + c) \end{equation}

Proofs

Proof 0

Let $f : \mathbb{C} \to \mathbb{C}$ be a D4312: Complex polynomial function such that

(i)	$a, b, c \in \mathbb{C}$ are each a D1207: Complex number
(ii)	\begin{equation} f(z) = a z^2 + (a b + a c) z + a b c \end{equation}

This result is a particular case of R4112: Strong fundamental theorem of complex algebra. $\square$