Whakaoti i te 12x^3+16x^2-41x-15

$$12 x ^ { 3 } + 16 x ^ { 2 } - 41 x - 15$$

Tauwehe

$$\left(2x-3\right)\left(2x+5\right)\left(3x+1\right)$$

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 -15, ā, ka wehea e q te whakarea arahanga 12. Ko tetahi pūtake pērā ko -\frac{1}{3}. Tauwehea te pūrau mā te whakawehe mā te 3x+1.

$$\left(3x+1\right)\left(4x^{2}+4x-15\right)$$

Whakaarohia te 4x^{2}+4x-15. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=4\left(-15\right)=-60$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-6 b=10$$

Tuhia anō te 4x^{2}+4x-15 hei \left(4x^{2}-6x\right)+\left(10x-15\right).

$$\left(4x^{2}-6x\right)+\left(10x-15\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(2x-3\right)+5\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(2x+5\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$\left(2x-3\right)\left(3x+1\right)\left(2x+5\right)$$

Aromātai

$$\left(2x-3\right)\left(2x+5\right)\left(3x+1\right)$$