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Beliebt Algebra >

z^6+4z^3+16=0

Lösung

z^{6} + 4 z^{3} + 16 = 0

Lösung

z = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), z = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), z = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9}), z = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), z = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), z = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

Schritte zur Lösung

z^{6} + 4 z^{3} + 16 = 0

Schreibe die Gleichung um mit

u = z^{3}

und

u^{2} = z^{6}

u^{2} + 4 u + 16 = 0

Löse

u^{2} + 4 u + 16 = 0 : u = - 2 + 23 i, u = - 2 - 23 i

u = - 2 + 23 i, u = - 2 - 23 i

Setze

u = z^{3} w i e d er e in,

löse für

z

Löse

z^{3} = - 2 + 23 i : z = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), z = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), z = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9})

Löse

z^{3} = - 2 - 23 i : z = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), z = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), z = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

Die Lösungen sind

z = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), z = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), z = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9}), z = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), z = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), z = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

Graph

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