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.