Whakaoti i te 3x^{2}+5x+7+2x^3+3x+16/x^2+x=10x^3+12x+4/x-2+7x^3/x+1

$$3 x ^ { 2 } + 5 x + 7 + \frac { 2 x ^ { 3 } + 3 x + 16 } { x ^ { 2 } + x } = \frac { 10 x ^ { 3 } + 12 x + 4 } { x } - \frac { 2 + 7 x ^ { 3 } } { x + 1 }$$

Whakaoti mō x

$$x=3$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+x,x,x+1.

$$3x^{2}x\left(x+1\right)+5xx\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 2 me te 1 kia riro ai te 3.

$$3x^{3}\left(x+1\right)+5xx\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{3} ki te x+1.

$$3x^{4}+3x^{3}+5xx\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$3x^{4}+3x^{3}+5x^{2}\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x^{2} ki te x+1.

$$3x^{4}+3x^{3}+5x^{3}+5x^{2}+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Pahekotia te 3x^{3} me 5x^{3}, ka 8x^{3}.

$$3x^{4}+8x^{3}+5x^{2}+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

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

$$3x^{4}+8x^{3}+5x^{2}+\left(x^{2}+x\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

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

$$3x^{4}+8x^{3}+5x^{2}+7x^{2}+7x+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Pahekotia te 5x^{2} me 7x^{2}, ka 12x^{2}.

$$3x^{4}+8x^{3}+12x^{2}+7x+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Pahekotia te 8x^{3} me 2x^{3}, ka 10x^{3}.

$$3x^{4}+10x^{3}+12x^{2}+7x+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Pahekotia te 7x me 3x, ka 10x.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te 10x^{3}+12x+4 ka whakakotahi i ngā kupu rite.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+16x+10x^{3}+4-x\left(2+7x^{3}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2+7x^{3}.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+16x+10x^{3}+4-\left(2x+7x^{4}\right)$$

Hei kimi i te tauaro o 2x+7x^{4}, kimihia te tauaro o ia taurangi.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+16x+10x^{3}+4-2x-7x^{4}$$

Pahekotia te 16x me -2x, ka 14x.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+14x+10x^{3}+4-7x^{4}$$

Pahekotia te 10x^{4} me -7x^{4}, ka 3x^{4}.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=3x^{4}+12x^{2}+14x+10x^{3}+4$$

Tangohia te 3x^{4} mai i ngā taha e rua.

$$3x^{4}+10x^{3}+12x^{2}+10x+16-3x^{4}=12x^{2}+14x+10x^{3}+4$$

Pahekotia te 3x^{4} me -3x^{4}, ka 0.

$$10x^{3}+12x^{2}+10x+16=12x^{2}+14x+10x^{3}+4$$

Tangohia te 12x^{2} mai i ngā taha e rua.

$$10x^{3}+12x^{2}+10x+16-12x^{2}=14x+10x^{3}+4$$

Pahekotia te 12x^{2} me -12x^{2}, ka 0.

$$10x^{3}+10x+16=14x+10x^{3}+4$$

Tangohia te 14x mai i ngā taha e rua.

$$10x^{3}+10x+16-14x=10x^{3}+4$$

Pahekotia te 10x me -14x, ka -4x.

$$10x^{3}-4x+16=10x^{3}+4$$

Tangohia te 10x^{3} mai i ngā taha e rua.

$$10x^{3}-4x+16-10x^{3}=4$$

Pahekotia te 10x^{3} me -10x^{3}, ka 0.

$$-4x+16=4$$

Tangohia te 16 mai i ngā taha e rua.

$$-4x=4-16$$

Tangohia te 16 i te 4, ka -12.

$$-4x=-12$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{-12}{-4}$$

Whakawehea te -12 ki te -4, kia riro ko 3.

$$x=3$$