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:
- $\ds \map P x = k \prod_{j \mathop = 1}^n \paren {x - \xi_j}$
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 $\zeta \in \C: \map P \zeta = 0$.
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:
- $\ds \map P z = k \prod_{j \mathop = 1}^n \paren {z - \zeta_j}$
where $k \in \C$.
Proof
Recall that Real Numbers form Field.
The result then follows from the Polynomial Factor Theorem.
$\blacksquare$
Also known as
Some sources refer to the Polynomial Factor Theorem (and its variants) as the Factor Theorem, but there are multiple theorems with this name.
Sources
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous): Chapter $1$: Polynomials; The Remainder and Factor Theorems; Undetermined Coefficients; Partial Fractions: $1.2$. The remainder and factor theorems
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 3$: Natural Numbers: Exercise $\S 3.11 \ (3)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor theorem