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Bevis for sætn.3 Faktorisering af 2.gradspolynomium 

Bevis for sætn.3 Faktorisering af 2.gradspolynomium .  Vides r 1 og r 2 er rødder i 2.gradspolynomiet p(x) = a x 2 + b x + c. Skal vises: p(x) = a(x –r 1 ) (x-r 2 ). Benyt definition af rødder og løsningsformlen til 2.gradsligningen. Kvadratsætning (x – q) (x + q) = x 2 – q 2.

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Bevis for sætn.3 Faktorisering af 2.gradspolynomium 

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  1. Bevis for sætn.3 Faktorisering af 2.gradspolynomium   Vides r1 og r2 er rødder i 2.gradspolynomiet p(x) = ax2 + bx + c Skal vises: p(x) = a(x –r1)(x-r2) Benyt definition af rødder og løsningsformlen til 2.gradsligningen Kvadratsætning (x – q)(x + q) = x2 – q2 og Kvadratsætning (x + q)2 = x2 + q2 + 2xq Indsæt d = b2 – 4ac Indsæt i højresiden af p(x) = a(x –r1)(x-r2) Sæt på fælles brøkstreg Gang a ind i parentesen

  2. Bevis for sætn.3 Faktorisering af 2.gradspolynomium   Vides p(x) = a(x –r1)(x-r2) Skal vises: r1 og r2 er rødder i 2.gradspolynomiet p(x) = ax2 + bx + c , dvs. at p(r1) = p(r2) = 0 Bestem hhv. p(r1) og p(r2) ved indsættelse p(r1) = a(r1-r1)(r1-r2) = a0(r1-r2) = 0 p(r2) = a(r2 -r1)(r2-r2) = a(r2-r1)0 = 0 r1 og r2 er altså rødder i p(x)

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