# Polynomial Factor Theorem/Corollary

## Corollary to Polynomial Factor Theorem

Let $\map P x$ be a polynomial in $x$ over the real numbers $\R$ of degree $n$.

Suppose there exists $\xi \in \R: \map P \xi = 0$.

Then $\map P x = \paren {x - \xi} \map Q x$, where $\map Q x$ is a polynomial of degree $n - 1$.

Hence, if $\xi_1, \xi_2, \dotsc, \xi_n \in \R$ such that all are different, and $\map P {\xi_1} = \map P {\xi_2} = \dotsb = \map P {\xi_n} = 0$, then:

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

where $k \in \R$.

### Complex Number form

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 Real Numbers form Field.

The result then follows from the Polynomial Factor Theorem.

$\blacksquare$

## Also known as

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

## Sources

- 1960: Margaret M. Gow:
*A Course in Pure Mathematics*... (previous): Chapter $1$: Polynomials; The Remainder and Factor Theorems; Undetermined Coefficients; Partial Fractions: $1.2$. The remainder and factor theorems - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 3$: Natural Numbers: Exercise $\S 3.11 \ (3)$