Polynomial Factor Theorem/Corollary

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

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

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

Hence, if $\xi_1, \xi_2, \ldots, \xi_n \in \R$ such that all are different, and $P \left({\xi_1}\right) = P \left({\xi_2}\right) = \ldots = P \left({\xi_n}\right) = 0$, then:
 * $\displaystyle P \left({x}\right) = k \prod_{j \mathop = 1}^n \left({x - \xi_j}\right)$

where $k \in \R$.

Proof
Recall that Real Numbers form Field.

The result then follows from the Polynomial Factor Theorem.