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

where $k \in \R$.

Proof
Recall that Real Numbers form Field.

The result then follows from the Polynomial Factor Theorem.

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.