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# zero of polynomial

Let $R$ be a subring of a commutative ring $S$. If $f$ is a polynomial in $R[X]$, it defines an evaluation homomorphism from $S$ to $S$. Any element $\alpha$ of $S$ satisfying

$f(\alpha)\;=\;0$ |

is a zero of the polynomial $f$.

If $R$ also is equipped with a non-zero unity, then the polynomial $f$ is in $S[X]$ divisible by the binomial
$X\!-\!\alpha$ (cf. the factor theorem). In this case, if $f$ is divisible by $(X\!-\!\alpha)^{n}$ but not by
$(X\!-\!\alpha)^{{n+1}}$, then $\alpha$ is a zero of the *order* $n$ of the polynomial $f$. If this order is 1, then $\alpha$ is a *simple zero* of $f$.

For example, the real number $\sqrt{2}$ ($\in\mathbb{R}$) is a zero of the polynomial $X^{2}\!-\!2$ of the polynomial ring $\mathbb{Q}[X]$.

Defines:

zero of polynomial, order of zero, order, simple zero, simple

Keywords:

zero, order, simple

Related:

PolynomialFunction, ZerosAndPolesOfRationalFunction

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

13P05*no label found*11C08

*no label found*12E05

*no label found*

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