x^{10}+x^5+1=0

Lösung

x^{10} + x^{5} + 1 = 0

x = cos (\frac{2 π}{15}) + i sin (\frac{2 π}{15}), x = cos (\frac{8 π}{15}) + i sin (\frac{8 π}{15}), x = cos (\frac{14 π}{15}) + i sin (\frac{14 π}{15}), x = - \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{26 π}{15}) + i sin (\frac{26 π}{15}), x = cos (\frac{2 π}{15}) - i sin (\frac{2 π}{15}), x = cos (\frac{4 π}{15}) + i sin (\frac{4 π}{15}), x = - \frac{1}{2} + i \frac{3}{2}, x = cos (\frac{16 π}{15}) + i sin (\frac{16 π}{15}), x = cos (\frac{22 π}{15}) + i sin (\frac{22 π}{15})

Schritte zur Lösung

x^{10} + x^{5} + 1 = 0

Schreibe die Gleichung um mit

u = x^{5}

und

u^{2} = x^{10}

u^{2} + u + 1 = 0

Löse

u^{2} + u + 1 = 0 : u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

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

Setze

u = x^{5} w i e d er e in,

löse für

x

Löse

x^{5} = - \frac{1}{2} + i \frac{3}{2} : x = cos (\frac{2 π}{15}) + i sin (\frac{2 π}{15}), x = cos (\frac{8 π}{15}) + i sin (\frac{8 π}{15}), x = cos (\frac{14 π}{15}) + i sin (\frac{14 π}{15}), x = - \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{26 π}{15}) + i sin (\frac{26 π}{15})

Löse

x^{5} = - \frac{1}{2} - i \frac{3}{2} : x = cos (\frac{2 π}{15}) - i sin (\frac{2 π}{15}), x = cos (\frac{4 π}{15}) + i sin (\frac{4 π}{15}), x = - \frac{1}{2} + i \frac{3}{2}, x = cos (\frac{16 π}{15}) + i sin (\frac{16 π}{15}), x = cos (\frac{22 π}{15}) + i sin (\frac{22 π}{15})

Die Lösungen sind

x = cos (\frac{2 π}{15}) + i sin (\frac{2 π}{15}), x = cos (\frac{8 π}{15}) + i sin (\frac{8 π}{15}), x = cos (\frac{14 π}{15}) + i sin (\frac{14 π}{15}), x = - \frac{1}{2} - i \frac{3}{2}, x = cos (\frac{26 π}{15}) + i sin (\frac{26 π}{15}), x = cos (\frac{2 π}{15}) - i sin (\frac{2 π}{15}), x = cos (\frac{4 π}{15}) + i sin (\frac{4 π}{15}), x = - \frac{1}{2} + i \frac{3}{2}, x = cos (\frac{16 π}{15}) + i sin (\frac{16 π}{15}), x = cos (\frac{22 π}{15}) + i sin (\frac{22 π}{15})

y^{3} - 25 y^{2} - 178 y + 1165 = 0

x^{4} - 3 = 2 x + 1

so l v e f or x, g = r (x + q)^{7} + s

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

x^{4} - 4 x^{3} + 6 x^{2} - 4 x - 1 = 0