# 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 $\mathbb{Q}[X]$.

 Title zero of polynomial Canonical name ZeroOfPolynomial Date of creation 2013-03-22 18:19:50 Last modified on 2013-03-22 18:19:50 Owner pahio (2872) Last modified by pahio (2872) Numerical id 8 Author pahio (2872) Entry type Definition Classification msc 13P05 Classification msc 11C08 Classification msc 12E05 Related topic PolynomialFunction Related topic ZerosAndPolesOfRationalFunction Defines zero of polynomial Defines order of zero Defines order Defines simple zero Defines simple