z^5+z=0

Lösung

z^{5} + z = 0

z = 0, z = - \frac{2}{2} + i \frac{2}{2}, z = - \frac{2}{2} - i \frac{2}{2}, z = \frac{2}{2} + i \frac{2}{2}, z = \frac{2}{2} - i \frac{2}{2}

Schritte zur Lösung

z^{5} + z = 0

Faktorisiere

z^{5} + z : z (z^{2} + 2 z + 1) (z^{2} - 2 z + 1)

z (z^{2} + 2 z + 1) (z^{2} - 2 z + 1) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

z = 0 or z^{2} + 2 z + 1 = 0 or z^{2} - 2 z + 1 = 0

Löse

z^{2} + 2 z + 1 = 0 : z = - \frac{2}{2} + i \frac{2}{2}, z = - \frac{2}{2} - i \frac{2}{2}

Löse

z^{2} - 2 z + 1 = 0 : z = \frac{2}{2} + i \frac{2}{2}, z = \frac{2}{2} - i \frac{2}{2}

Die Lösungen sind

z = 0, z = - \frac{2}{2} + i \frac{2}{2}, z = - \frac{2}{2} - i \frac{2}{2}, z = \frac{2}{2} + i \frac{2}{2}, z = \frac{2}{2} - i \frac{2}{2}

5 x^{3} + 5102 x + 2015 = 0

5 x - 6 = 2 x^{3} - 3 x^{2}

(2 + b) (11 b - 2 b^{2}) + 189 = 0

so l v e f or x, x^{4} y^{3} - 2 x y^{4} = x y^{3}

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