Polynomial_Factor_Theorem

Let $P \paren x$ be a polynomial in $x$ over a field $K$ of degree $n$. Then $\xi \in K: P \paren \xi = 0 \iff \map P x = \paren {x - \xi} \map Q x$ where $Q$ is a polynomial of degree $n - 1$. Hence, if $\xi_1, \xi_2, \ldots, \xi_n \in K$ 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 $k \in K$.