z^6-(3+2i)z^3+2+2i=0

Lösung

z^{6} - (3 + 2 i) z^{3} + 2 + 2 i = 0

z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = \frac{1 + 3}{2} + i \frac{- 1 + 3}{2}, z = - 1 + i, z = \frac{1 - 3}{2} + i \frac{- 1 - 3}{2}

Schritte zur Lösung

z^{6} - (3 + 2 i) z^{3} + 2 + 2 i = 0

Fasse zusammen

(z^{6} - 3 z^{3} + 2) + i (2 - 2 z^{3}) = 0

Schreibe in der Standard Form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

z^{6} - (3 + 2 i) z^{3} + 2 + 2 i = 0

Schreibe die Gleichung um mit

u = z^{3}

und

u^{2} = z^{6}

u^{2} - (3 + 2 i) u + 2 + 2 i = 0

Löse

u^{2} - (3 + 2 i) u + 2 + 2 i = 0 : u = 1, u = 2 + 2 i

u = 1, u = 2 + 2 i

Setze

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

löse für

z

Löse

z^{3} = 1 : z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i

Löse

z^{3} = 2 + 2 i : z = \frac{1 + 3}{2} + i \frac{- 1 + 3}{2}, z = - 1 + i, z = \frac{1 - 3}{2} + i \frac{- 1 - 3}{2}

Die Lösungen sind

z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = \frac{1 + 3}{2} + i \frac{- 1 + 3}{2}, z = - 1 + i, z = \frac{1 - 3}{2} + i \frac{- 1 - 3}{2}

roo t s x^{3} - 3 x^{2} - 10 x + 24

x^{3} - x^{2} - 25 x + 25 = 0

3 y^{3} - 20 y^{2} - 7 y = 0

4 x^{3} - 5 x + 2 = 24

2 x^{3} - 6 x^{2} + 13 x - 14 = 0