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Vil du være god til polynomier? Lær alt hvad du skal kunne om polynomier og andengradspolynomier på under 5 minutter!

Polynomier

Langt de fleste elever er igennem deres skoletid stødt på polynomier, men dette skal ikke forhindre os i at lave en kort gennemgang af hvad polynomier egentlig er.

Et polynomium er egentlig bare en matematisk funktion, som følger en bestemt forskrift. Vi vil starte med at opskrive udtrykket for et såkaldt n’tegradspolynomium. Et n’tegradspolynomium er givet ved udtrykket

Dette udtryk kan måske virke en smule intimiderende, men der en mening med galskaben.

 

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Tallet n angiver graden af vores polynomium. Er n=5, så vil vores polynomium være et femtegradspolynomium og have formen

Hvor a5, a4, a3, a2, a1 og a0 er konstanter. Har konstanterne en værdi kunne vores femtegradspolynomium have formen

Forneden ses grafen for ovenstående polynomie.

Læg mærke til, at det er den højeste potens, som der angiver graden af polynomiet, derved kunne et femtegradspolynomium blot have formen

Lad os nu betragte det simpleste polynomium vi kan skrive, nemlig et førstegradspolynomium

Den opmærksomme læser vil her kunne se at, der blot er tale om en lineær funktion der har hældningen a1 og som skærer andenaksen i y = a0.

Et eksempel på et førstegradspolynomie kunne for eksempel være

Forneden ses grafen for førstegradspolynomiet og her ses det tydeligt at et førstegradspolynomie blot er en ret linje

En af de mest populære polynomier som du helt sikkert har stiftet bekendskab til er et andengradspolynomie. Et andengradspolynomium har formen

Et eksempel på et andengradspolynomium kunne være

Forneden ses grafen for andengradspolynomiet

 

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Som du sikkert allerede ved så kaldes grafen for et andengradspolynomium for en parabel. En parabel består af et toppunkt og to grene. I de fleste matematikbøger er andengradspolynomium oftest skrevet på formen

Hvis a > 0, så vender parablens grene og parablens åbning opad, denne parabel kaldes oftest en glad parabel. Når man har en glad parabel, så er parablens toppunkt et minimumspunkt.

Er a < 0, så vender parablens grene nedad. Denne parabel kaldes ofte en sur parabel. Når man har med en sur parabel at gøre, så er parablens toppunkt et maksimumspunkt.

Hvordan kan man så bestemme en parabels toppunkt?

Har man et andengradspolynomium med formen

er toppunktet hos funktionens tilhørende parabel givet ved

Hvor d angiver polynomiets diskriminant som er givet ved

Vi kan desuden bestemme parablens nulpunkter, dvs. f(x) = 0.

Dette gøres ved at løse andengradsligningen

Her er løsningen givet ved

Læg mærke til at hvis diskriminanten d > 0, så har parablen to nulpunkter.  Er d = 0, så har den et nulpunkt, og er d < 0, så har parablen ingen nulpunkter.

En tommelfingerregel er at et andengradspolynomium kan have to løsninger, et tredjegradspolynomium kan have op til tre løsninger osv.

Dette var en lille gennemgang af polynomier, med det mest essentielle. Nu er der vist ikke andet for end at gå i gang med at regne en masse opgaver 🙂

Jeg giver dig samtidigt nogle gode links til andre spænende artikler

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