(2-i)/(1+2i)*z^4=1

Lösung

\frac{2 - i}{1 + 2 i} \cdot z^{4} = 1

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

Schritte zur Lösung

\frac{2 - i}{1 + 2 i} z^{4} = 1

Vereinfache

\frac{2 - i}{1 + 2 i} : - i

(- i) z^{4} = 1

Teile beide Seiten durch

- i

z^{4} = i

Für

z^{n} = a

sind die Lösungen

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

Für

n = 4, a = i

∣ a ∣ = 1

ar g (a) = \frac{π}{2}

z = 41 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{4})), z = 41 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{4})), z = 41 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{4})), z = 41 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{4}))

Vereinfache

41 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{4})) : \frac{2 + 2}{2} + i \frac{2 - 2}{2}

Vereinfache

41 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{4})) : - \frac{2 - 2}{2} + i \frac{2 + 2}{2}

Vereinfache

41 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{4})) : - \frac{2 + 2}{2} - i \frac{2 - 2}{2}

Vereinfache

41 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{4})) : \frac{2 - 2}{2} - i \frac{2 + 2}{2}

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

x^{3} + x^{2} + 2 x + 1 = 0

x^{3} - 40 x^{2} + 404 x - 80 = 0

(x + 2) (x + 3) (x + 8) (x + 12) = 54 x^{2}

8 x^{3} + 64 x^{2} + 120 x = 0

x^{3} - x^{2} - 4 x + 8 = 0