Whakaoti i te 1/x^2+x-2+1/x^2+7x+10+1/x^2+13x+40=1/3x-3-frac{1}{21}

$$\frac { 1 } { x ^ { 2 } + x - 2 } + \frac { 1 } { x ^ { 2 } + 7 x + 10 } + \frac { 1 } { x ^ { 2 } + 13 x + 40 } = \frac { 1 } { 3 x - 3 } - \frac { 1 } { 21 }$$

Whakaoti mō x

$$x=-1$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -8,-5,-2,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+x-2,x^{2}+7x+10,x^{2}+13x+40,3x-3,21.

$$21\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 21 ki te x+5.

$$\left(21x+105\right)\left(x+8\right)+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 21x+105 ki te x+8 ka whakakotahi i ngā kupu rite.

$$21x^{2}+273x+840+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 21 ki te x-1.

$$21x^{2}+273x+840+\left(21x-21\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 21x-21 ki te x+8 ka whakakotahi i ngā kupu rite.

$$21x^{2}+273x+840+21x^{2}+147x-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Pahekotia te 21x^{2} me 21x^{2}, ka 42x^{2}.

$$42x^{2}+273x+840+147x-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Pahekotia te 273x me 147x, ka 420x.

$$42x^{2}+420x+840-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Tangohia te 168 i te 840, ka 672.

$$42x^{2}+420x+672+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 21 ki te x+2.

$$42x^{2}+420x+672+\left(21x+42\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 21x+42 ki te x-1 ka whakakotahi i ngā kupu rite.

$$42x^{2}+420x+672+21x^{2}+21x-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Pahekotia te 42x^{2} me 21x^{2}, ka 63x^{2}.

$$63x^{2}+420x+672+21x-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Pahekotia te 420x me 21x, ka 441x.

$$63x^{2}+441x+672-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Tangohia te 42 i te 672, ka 630.

$$63x^{2}+441x+630=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x+2.

$$63x^{2}+441x+630=\left(7x+14\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 7x+14 ki te x+5 ka whakakotahi i ngā kupu rite.

$$63x^{2}+441x+630=\left(7x^{2}+49x+70\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 7x^{2}+49x+70 ki te x+8 ka whakakotahi i ngā kupu rite.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

Whakareatia te 21 ki te -\frac{1}{21}, ka -1.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560-\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x-1.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x+1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x+1 ki te x+2 ka whakakotahi i ngā kupu rite.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x^{2}-x+2\right)\left(x+5\right)\left(x+8\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x^{2}-x+2 ki te x+5 ka whakakotahi i ngā kupu rite.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x^{3}-6x^{2}-3x+10\right)\left(x+8\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x^{3}-6x^{2}-3x+10 ki te x+8 ka whakakotahi i ngā kupu rite.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560-x^{4}-14x^{3}-51x^{2}-14x+80$$

Pahekotia te 7x^{3} me -14x^{3}, ka -7x^{3}.

$$63x^{2}+441x+630=-7x^{3}+105x^{2}+462x+560-x^{4}-51x^{2}-14x+80$$

Pahekotia te 105x^{2} me -51x^{2}, ka 54x^{2}.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+462x+560-x^{4}-14x+80$$

Pahekotia te 462x me -14x, ka 448x.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+448x+560-x^{4}+80$$

Tāpirihia te 560 ki te 80, ka 640.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+448x+640-x^{4}$$

Me tāpiri te 7x^{3} ki ngā taha e rua.

$$63x^{2}+441x+630+7x^{3}=54x^{2}+448x+640-x^{4}$$

Tangohia te 54x^{2} mai i ngā taha e rua.

$$63x^{2}+441x+630+7x^{3}-54x^{2}=448x+640-x^{4}$$

Pahekotia te 63x^{2} me -54x^{2}, ka 9x^{2}.

$$9x^{2}+441x+630+7x^{3}=448x+640-x^{4}$$

Tangohia te 448x mai i ngā taha e rua.

$$9x^{2}+441x+630+7x^{3}-448x=640-x^{4}$$

Pahekotia te 441x me -448x, ka -7x.

$$9x^{2}-7x+630+7x^{3}=640-x^{4}$$

Tangohia te 640 mai i ngā taha e rua.

$$9x^{2}-7x+630+7x^{3}-640=-x^{4}$$

Tangohia te 640 i te 630, ka -10.

$$9x^{2}-7x-10+7x^{3}=-x^{4}$$

Me tāpiri te x^{4} ki ngā taha e rua.

$$9x^{2}-7x-10+7x^{3}+x^{4}=0$$

Hurinahatia te whārite ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{4}+7x^{3}+9x^{2}-7x-10=0$$

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 -10, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±10,±5,±2,±1$$

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^{4}+7x^{3}+9x^{2}-7x-10 ki te x-1, kia riro ko x^{3}+8x^{2}+17x+10. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{3}+8x^{2}+17x+10=0$$

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

$$±10,±5,±2,±1$$

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^{3}+8x^{2}+17x+10 ki te x+1, kia riro ko x^{2}+7x+10. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{2}+7x+10=0$$

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 7 mō te b, me te 10 mō te c i te ture pūrua.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\times 10}}{2}$$

Mahia ngā tātaitai.

$$x=\frac{-7±3}{2}$$

Whakaotia te whārite x^{2}+7x+10=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=-5 x=-2$$

Tangohia ngā uara e kore e ōrite ki te taurangi.

$$x=-1$$

Rārangitia ngā otinga katoa i kitea.

$$x=1 x=-1 x=-5 x=-2$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 1,-5,-2.

$$x=-1$$