Binomisk Formeln

A Binomisk Formeln san en wai uun a matematiik, am det moolnemen faan sumen ianfacher tu maagin. Jo halep uk bi't bröögreegnin mä sumen.

Tekst üüb Öömrang

Det wurd binomisk komt faan bi (tau) an Nomen (nööm).

Formeln

• Algemian

${\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}\cdot b^{k}}$ , ${\displaystyle n\in \mathbb {N} }$

• Enkelt

${\displaystyle (a+b)^{2}=a^{2}+2\cdot a\cdot b+b^{2}}$	iarst Binomisk Formel (Plus-Formel)
${\displaystyle (a-b)^{2}=a^{2}-2\cdot a\cdot b+b^{2}}$	ööder Binomisk Formel (Minus-Formel)
${\displaystyle (a+b)\cdot (a-b)=a^{2}-b^{2}}$	traad Binomisk Formel (Plus-Minus-Formel)

Troch moolnemen koon am a formeln bewise:

${\displaystyle (a+b)^{2}=(a+b)\cdot (a+b)=a\cdot a+a\cdot b+b\cdot a+b\cdot b=a^{2}+2\cdot a\cdot b+b^{2}}$

${\displaystyle (a-b)^{2}=(a-b)\cdot (a-b)=a\cdot a-a\cdot b-b\cdot a+b\cdot b=a^{2}-2\cdot a\cdot b+b^{2}}$

${\displaystyle (a+b)\cdot (a-b)=a\cdot a-a\cdot b+b\cdot a-b\cdot b=a^{2}-b^{2}}$

A formeln halep uk bi't hoodreegnin:

Bispal 1

${\displaystyle 37^{2}=(30+7)^{2}=30^{2}+2\cdot 30\cdot 7+7^{2}=900+420+49=1369}$

Bispal 2

${\displaystyle 17\cdot 13=(15+2)\cdot (15-2)=15^{2}-2^{2}=225-4=221}$

Bewis uun a geometrii

	Det Kwadroot as (a+b) lung an briad. Jo letj kwadrooten a² an b² paase diar jüst iin, an tau likedenang rochthuken a·b bliiw auer. Sodenang as ${\displaystyle (a+b)^{2}=a^{2}+2\cdot a\cdot b+b^{2}}$
	Uun't ööder bil as a² det blä kwadroot. Wan diarütj det kwadroot wurd skal, wat (a-b) lung an briad as, namst dü jo tau ruad rochthuken wech, diar a·b grat san. Nü heest dü diarmä oober b² tweisis wechnimen, diaram skal am det iansis weder diartutääl. Sodenang as ${\displaystyle (a-b)^{2}=a^{2}-2\cdot a\cdot b+b^{2}}$
	Uun't traad bil as a² det laacht- an jonkblä kwadroot. Woort det letj kwadroot b² wechnimen, an det rochthuk, wat auerblaft, oner bihinget, jaft det en rochthuk, wat (a-b) briad an (a+b) huuch as. Sodenang as ${\displaystyle a^{2}-b^{2}=(a+b)\cdot (a-b)}$

Luke uk diar

 Commonskategorii: Binomisk Formeln – Saamlang faan bilen of filmer