Polynomial Factor Theorem/Corollary

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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 $k \in \R$.


Complex Number form

Let $\map P z$ be a polynomial in $z$ over the complex numbers $\C$ of degree $n$.

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


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:

$\displaystyle \map P z = k \prod_{j \mathop = 1}^n \paren {z - \zeta_j}$

where $k \in \C$.


Proof

Recall that Real Numbers form Field.

The result then follows from the Polynomial Factor Theorem.

$\blacksquare$


Sources