Whakaoti i te z^6-z^3+1=0

$$z ^ { 6 } - z ^ { 3 } + 1 = 0$$

Whakaoti mō z

$$z\in e^{\frac{5\pi i}{9}},e^{\frac{17\pi i}{9}},e^{\frac{11\pi i}{9}},e^{\frac{13\pi i}{9}},e^{\frac{\pi i}{9}},e^{\frac{7\pi i}{9}}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakakapia te t mō te z^{3}.

$$t^{2}-t+1=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 1 mō te a, te -1 mō te b, me te 1 mō te c i te ture pūrua.

$$t=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 1}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{1±\sqrt{-3}}{2}$$

Whakaotia te whārite t=\frac{1±\sqrt{-3}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=\frac{1+\sqrt{3}i}{2} t=\frac{-\sqrt{3}i+1}{2}$$

Mai i te z=t^{3}, ka taea ngā otinga mā te whakaoti te whārite mō ia t.

$$z=-e^{\frac{4\pi i}{9}} z=ie^{\frac{5\pi i}{18}} z=e^{\frac{\pi i}{9}} z=-ie^{\frac{7\pi i}{18}} z=-e^{\frac{2\pi i}{9}} z=ie^{\frac{\pi i}{18}}$$