Whakaoti i te x^3+2x^2-41x-42

$$x ^ { 3 } + 2 x ^ { 2 } - 41 x - 42$$

Tauwehe

$$\left(x-6\right)\left(x+1\right)\left(x+7\right)$$

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 -42, ā, ka wehea e q te whakarea arahanga 1. Ko tetahi pūtake pērā ko -7. Tauwehea te pūrau mā te whakawehe mā te x+7.

$$\left(x+7\right)\left(x^{2}-5x-6\right)$$

Whakaarohia te x^{2}-5x-6. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=1\left(-6\right)=-6$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -6.

$$1,-6 2,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-6=-5 2-3=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-6 b=1$$

Tuhia anō te x^{2}-5x-6 hei \left(x^{2}-6x\right)+\left(x-6\right).

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

Whakatauwehea atu x i te x^{2}-6x.

$$x\left(x-6\right)+x-6$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x+1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$\left(x-6\right)\left(x+1\right)\left(x+7\right)$$

Aromātai

$$\left(x-6\right)\left(x+1\right)\left(x+7\right)$$