Whakaoti i te z^3-1=0

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

Whakaoti mō z (complex solution)

$$z=\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

$$z=1$$

$$z=\frac{-1+\sqrt{3}i}{2}\approx -0.5+0.866025404i$$

Ngā Upane Otinga

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -1, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$z=1$$

Mā te whakatakotoranga Tauwehe, he tauwehe te z-k o te pūrau mō ia pūtake k. Whakawehea te z^{3}-1 ki te z-1, kia riro ko z^{2}+z+1. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$z^{2}+z+1=0$$

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.

$$z=\frac{-1±\sqrt{1^{2}-4\times 1\times 1}}{2}$$

Mahia ngā tātaitai.

$$z=\frac{-1±\sqrt{-3}}{2}$$

Whakaotia te whārite z^{2}+z+1=0 ina he tōrunga te ±, ina he tōraro te ±.

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

Rārangitia ngā otinga katoa i kitea.

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

Whakaoti mō z

$$z=1$$

Ngā Upane Otinga

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -1, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$z=1$$

Mā te whakatakotoranga Tauwehe, he tauwehe te z-k o te pūrau mō ia pūtake k. Whakawehea te z^{3}-1 ki te z-1, kia riro ko z^{2}+z+1. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$z^{2}+z+1=0$$

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.

$$z=\frac{-1±\sqrt{1^{2}-4\times 1\times 1}}{2}$$

Mahia ngā tātaitai.

$$z=\frac{-1±\sqrt{-3}}{2}$$

Tā te mea e kore te pūrua o tētahi tau tōraro e tautohutia ki te āpure tūturu, kāhore he rongoā.

$$z\in \emptyset$$

Rārangitia ngā otinga katoa i kitea.

$$z=1$$