Whakaoti i te 3x^4+24x^3+48x^2-x^2-8x-16

$$3 x ^ { 4 } + 24 x ^ { 3 } + 48 x ^ { 2 } - x ^ { 2 } - 8 x - 16$$

Tauwehe

$$\left(3x^{2}-1\right)\left(x+4\right)^{2}$$

Ngā Upane Otinga

Whakarea ka paheko i ngā kīanga tau ōrite.

$$3x^{4}+24x^{3}+47x^{2}-8x-16$$

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

$$3x^{4}+24x^{3}+47x^{2}-8x-16=0$$

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 -16, ā, ka wehea e q te whakarea arahanga 3. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{16}{3},±16,±\frac{8}{3},±8,±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=-4$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 3x^{4}+24x^{3}+47x^{2}-8x-16 ki te x+4, kia riro ko 3x^{3}+12x^{2}-x-4. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$3x^{3}+12x^{2}-x-4=0$$

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 -4, ā, ka wehea e q te whakarea arahanga 3. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=-4$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 3x^{3}+12x^{2}-x-4 ki te x+4, kia riro ko 3x^{2}-1. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$3x^{2}-1=0$$

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 3 mō te a, te 0 mō te b, me te -1 mō te c i te ture pūrua.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Mahia ngā tātaitai.

$$x=\frac{0±2\sqrt{3}}{6}$$

Whakaotia te whārite 3x^{2}-1=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=-\frac{\sqrt{3}}{3} x=\frac{\sqrt{3}}{3}$$

Me tuhi anō te kīanga whakatauwehe mā ngā pūtake i riro. Kāore te pūrau 3x^{2}-1 i whakatauwehea i te mea kāhore ōna pūtake whakahau.

$$\left(3x^{2}-1\right)\left(x+4\right)^{2}$$

Aromātai

$$\left(3x^{2}-1\right)\left(x+4\right)^{2}$$

Ngā Upane Otinga

Pahekotia te 48x^{2} me -x^{2}, ka 47x^{2}.

$$3x^{4}+24x^{3}+47x^{2}-8x-16$$