Whakaoti i te (x-4)^2*(x+3)^3*(x-1)=0

$$( x - 4 ) ^ { 2 } \cdot ( x + 3 ) ^ { 3 } \cdot ( x - 1 ) = 0$$

Whakaoti mō x

$$x=-3$$

$$x=4$$

$$x=1$$

Ngā Upane Otinga

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-4\right)^{2}.

$$\left(x^{2}-8x+16\right)\left(x+3\right)^{3}\left(x-1\right)=0$$

Whakamahia te ture huarua \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} hei whakaroha \left(x+3\right)^{3}.

$$\left(x^{2}-8x+16\right)\left(x^{3}+9x^{2}+27x+27\right)\left(x-1\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x^{2}-8x+16 ki te x^{3}+9x^{2}+27x+27 ka whakakotahi i ngā kupu rite.

$$\left(x^{5}+x^{4}-29x^{3}-45x^{2}+216x+432\right)\left(x-1\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x^{5}+x^{4}-29x^{3}-45x^{2}+216x+432 ki te x-1 ka whakakotahi i ngā kupu rite.

$$x^{6}-30x^{4}-16x^{3}+261x^{2}+216x-432=0$$

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 -432, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±432,±216,±144,±108,±72,±54,±48,±36,±27,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

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.

$$x=1$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{6}-30x^{4}-16x^{3}+261x^{2}+216x-432 ki te x-1, kia riro ko x^{5}+x^{4}-29x^{3}-45x^{2}+216x+432. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{5}+x^{4}-29x^{3}-45x^{2}+216x+432=0$$

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

$$±432,±216,±144,±108,±72,±54,±48,±36,±27,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

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^{5}+x^{4}-29x^{3}-45x^{2}+216x+432 ki te x+3, kia riro ko x^{4}-2x^{3}-23x^{2}+24x+144. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{4}-2x^{3}-23x^{2}+24x+144=0$$

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

$$±144,±72,±48,±36,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

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}-2x^{3}-23x^{2}+24x+144 ki te x+3, kia riro ko x^{3}-5x^{2}-8x+48. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{3}-5x^{2}-8x+48=0$$

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

$$±48,±24,±16,±12,±8,±6,±4,±3,±2,±1$$

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}-5x^{2}-8x+48 ki te x+3, kia riro ko x^{2}-8x+16. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{2}-8x+16=0$$

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 -8 mō te b, me te 16 mō te c i te ture pūrua.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\times 16}}{2}$$

Mahia ngā tātaitai.

$$x=\frac{8±0}{2}$$

He ōrite ngā whakatau.

$$x=4$$

Rārangitia ngā otinga katoa i kitea.

$$x=1 x=-3 x=4$$