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Universität/Hochschule Irreduzibilität eines Polynoms
Math_user
Aktiv Letzter Besuch: in der letzten Woche
Dabei seit: 04.05.2019
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Guten Morgen zusammen

Wie kann ich auf eine möglichst einfache Weise zeigen, dass \(x^7-2x+2\in \mathbb{R} \) reduzibel, beziehungsweise irreduzibel?

Vielen Dank und einen guten Start in den Tag

Math_user



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xiao_shi_tou_
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2019-11-21 00:48 - Math_user im Themenstart schreibt:
Guten Morgen zusammen

Wie kann ich auf eine möglichst einfache Weise zeigen, dass \(x^7-2x+2\in \mathbb{R} \) reduzibel, beziehungsweise irreduzibel?

Vielen Dank und einen guten Start in den Tag

Math_user
Das Polynom hat einen ungeraden Grad und besitzt daher eine Nullstelle $\a$ in $\R$. Daher ist $x-\a$ ein Teiler des Polynoms, es gibt also $g\in \R[x]$ mit $x^7-2x+2=g(x)(x-\a)$. Da das Polynom vom Grad $>1$ ist kann $g(x)$ keine Einheit sein. Also ist das Polynom reduzibel.



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”己所不欲,勿施于人“(Konfuzius)
PS: Falls ich plötzlich aufhöre in einem Thread zu antworten, dann kann es sein, dass ich es vergessen habe. Ihr könnt mir in diesem Fall eine Private Nachricht schicken.
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