Polynomial Factor Theorem/Corollary/Complex Numbers

Corollary to Polynomial Factor Theorem
Let $P \left({z}\right)$ be a polynomial in $z$ over the complex numbers $\C$ of degree $n$.

Suppose there exists $\zeta \in \C: P \left({\xi}\right) = 0$.

Then $P \left({z}\right) = \left({x - \zeta}\right) Q \left({z}\right)$, where $Q \left({z}\right)$ is a polynomial of degree $n - 1$.

Hence, if $\zeta_1, \zeta_2, \ldots, \zeta_n \in \C$ such that all are different, and $P \left({\zeta_1}\right) = P \left({\zeta_2}\right) = \ldots = P \left({\zeta_n}\right) = 0$, then:
 * $\displaystyle P \left({z}\right) = k \prod_{j \mathop = 1}^n \left({z - \zeta_j}\right)$

where $k \in \C$.

Proof
Recall that Complex Numbers form Field.

The result then follows from the Polynomial Factor Theorem.