Polynomial Factor Theorem/Corollary/Complex Numbers

Corollary to Polynomial Factor Theorem

Let $\map P z$ be a polynomial in $z$ over the complex numbers $\C$ of degree $n$.

Suppose there exists $\zeta \in \C: \map P \zeta = 0$.

Then:

$\map P z = \paren {x - \zeta} \map Q z$

where $\map Q z$ is a polynomial of degree $n - 1$.

Hence, if $\zeta_1, \zeta_2, \ldots, \zeta_n \in \C$ such that all are different, and $\map P {\zeta_1} = \map P {\zeta_2} = \dotsb = \map P {\zeta_n} = 0$, then:

$\ds \map P z = k \prod_{j \mathop = 1}^n \paren {z - \zeta_j}$

where $k \in \C$.

Proof

Recall that Complex Numbers form Field.

The result then follows from the Polynomial Factor Theorem.

$\blacksquare$

Also known as

Some sources refer to the Polynomial Factor Theorem (and its variants) as the Factor Theorem, but there are multiple theorems with this name.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity