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Beliebt Algebra >

faktorisieren (a^2-1)^3-27

Lösung

faktorisieren

(a^{2} - 1)^{3} - 27

Lösung

(a + 2) (a - 2) (a^{4} + a^{2} + 7)

Schritte zur Lösung

(a^{2} - 1)^{3} - 27

Schreibe

27

um:

3^{3}

= (a^{2} - 1)^{3} - 3^{3}

Wende Formel zur Differenz von dritten Potenzen an:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(a^{2} - 1)^{3} - 3^{3} = ((a^{2} - 1) - 3) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 3^{2})

= ((a^{2} - 1) - 3) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 3^{2})

Fasse zusammen

= (a^{2} - 4) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 9)

Faktorisiere

a^{2} - 4 : (a + 2) (a - 2)

= (a + 2) (a - 2) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 9)

(a^{2} - 1)^{2} = (a + 1)^{2} (a - 1)^{2}

3 (a^{2} - 1) = 3 (a + 1) (a - 1)

= (a + 2) (a - 2) ((a + 1)^{2} (a - 1)^{2} + 3 (a + 1) (a - 1) + 9)

Multipliziere aus

(a + 1)^{2} (a - 1)^{2} + 3 (a + 1) (a - 1) + 9 : a^{4} + a^{2} + 7

= (a + 2) (a - 2) (a^{4} + a^{2} + 7)

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