Whakaoti i te x^4-18x^2+81

$${ x }^{ 4 } -18 { x }^{ 2 } +81$$

Tauwehe

$$\left(x-3\right)^{2}\left(x+3\right)^{2}$$

Ngā Upane Otinga

Kia tauwehea ai te kīanga, me whakaoti te whārite ina ōrite ki te 0.

$$x^{4}-18x^{2}+81=0$$

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 81, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±81,±27,±9,±3,±1$$

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.

$$x=3$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{4}-18x^{2}+81 ki te x-3, kia riro ko x^{3}+3x^{2}-9x-27. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$x^{3}+3x^{2}-9x-27=0$$

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 -27, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±27,±9,±3,±1$$

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.

$$x=3$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{3}+3x^{2}-9x-27 ki te x-3, kia riro ko x^{2}+6x+9. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$x^{2}+6x+9=0$$

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 6 mō te b, me te 9 mō te c i te ture pūrua.

$$x=\frac{-6±\sqrt{6^{2}-4\times 1\times 9}}{2}$$

Mahia ngā tātaitai.

$$x=\frac{-6±0}{2}$$

He ōrite ngā whakatau.

$$x=-3$$

Me tuhi anō te kīanga whakatauwehe mā ngā pūtake i riro.

$$\left(x-3\right)^{2}\left(x+3\right)^{2}$$

Aromātai

$$\left(x^{2}-9\right)^{2}$$