x^6-x^3+1=0

Lösung

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

x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

Schritte zur Lösung

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

Schreibe die Gleichung um mit

u = x^{3}

und

u^{2} = x^{6}

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^{3} w i e d er e in,

löse für

x

Löse

x^{3} = \frac{1}{2} + i \frac{3}{2} : x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9})

Löse

x^{3} = \frac{1}{2} - i \frac{3}{2} : x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

Die Lösungen sind

x = cos (\frac{π}{9}) + i sin (\frac{π}{9}), x = cos (\frac{7 π}{9}) + i sin (\frac{7 π}{9}), x = cos (\frac{13 π}{9}) + i sin (\frac{13 π}{9}), x = cos (\frac{π}{9}) - i sin (\frac{π}{9}), x = cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9}), x = cos (\frac{11 π}{9}) + i sin (\frac{11 π}{9})

z^{4} = 82 + 82 i

x^{3} - 7 x^{2} + 4 x + 30 = 0

(x) = x^{3} - 3 x^{2} - 9 x - 5

so l v e f or x, x^{2} + (y - c b r t (x^{2}))^{2} = 25

roo t s z^{4} - 3 z^{3} - 5 z^{2} + 6 z - 8