Løs (2x^2-1/(x-2)(x+1))^2-8(2x^2-1)+7(x-1)(x+2)

$$( 2 x ^ { 2 } - \frac { 1 } { ( x - 2 ) ( x + 1 ) } ) ^ { 2 } - 8 ( 2 x ^ { 2 } - 1 ) + 7 ( x - 1 ) ( x + 2 )$$

Evaluer

$$\frac{4x^{8}-8x^{7}-21x^{6}+41x^{5}+19x^{4}-41x^{3}+18x^{2}+4x-23}{\left(\left(x-2\right)\left(x+1\right)\right)^{2}}$$

Korte løsningstrinn

Hvis du vil legge til eller trekke fra uttrykk, kan du utvide dem for å gjøre nevnerne like. Multipliser 2x^{2} ganger \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}.

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

Siden \frac{2x^{2}\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} og \frac{1}{\left(x-2\right)\left(x+1\right)} har samme nevner, kan du subtrahere dem ved å subtrahere tellerne.

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

Utfør multiplikasjonene i 2x^{2}\left(x-2\right)\left(x+1\right)-1.

$$\left(\frac{2x^{4}+2x^{3}-4x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)}\right)^{2}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Kombiner like ledd i 2x^{4}+2x^{3}-4x^{3}-4x^{2}-1.

$$\left(\frac{2x^{4}-2x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)}\right)^{2}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Hvis du vil heve \frac{2x^{4}-2x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)} i en potens, øker du både telleren og nevneren i en potens, og deler deretter.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(\left(x-2\right)\left(x+1\right)\right)^{2}}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Utvid \left(\left(x-2\right)\left(x+1\right)\right)^{2}.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Bruk den distributive lov til å multiplisere -8 med 2x^{2}-1.

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

Bruk den distributive lov til å multiplisere 7 med x-1.

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

Bruk den distributive lov til å multiplisere 7x-7 med x+2 og kombinere like ledd.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-16x^{2}+8+7x^{2}+7x-14$$

Kombiner -16x^{2} og 7x^{2} for å få -9x^{2}.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-9x^{2}+8+7x-14$$

Trekk fra 14 fra 8 for å få -6.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-9x^{2}-6+7x$$

Hvis du vil legge til eller trekke fra uttrykk, kan du utvide dem for å gjøre nevnerne like. Multipliser -9x^{2}-6+7x ganger \frac{\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}+\frac{\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Siden \frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}} og \frac{\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}} har samme nevner, kan du legge dem sammen ved å legge sammen tellerne.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}+\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Utfør multiplikasjonene i \left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}+\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}.

$$\frac{4x^{8}-4x^{7}-8x^{6}-2x^{4}-4x^{7}+4x^{6}+8x^{5}+2x^{3}-8x^{6}+8x^{5}+16x^{4}+4x^{2}-2x^{4}+2x^{3}+4x^{2}+1-9x^{6}+18x^{5}+27x^{4}-36x^{3}-36x^{2}-6x^{4}+12x^{3}+18x^{2}-24x-24+7x^{5}-14x^{4}-21x^{3}+28x^{2}+28x}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Kombiner like ledd i 4x^{8}-4x^{7}-8x^{6}-2x^{4}-4x^{7}+4x^{6}+8x^{5}+2x^{3}-8x^{6}+8x^{5}+16x^{4}+4x^{2}-2x^{4}+2x^{3}+4x^{2}+1-9x^{6}+18x^{5}+27x^{4}-36x^{3}-36x^{2}-6x^{4}+12x^{3}+18x^{2}-24x-24+7x^{5}-14x^{4}-21x^{3}+28x^{2}+28x.

$$\frac{4x^{8}-8x^{7}-21x^{6}+19x^{4}+41x^{5}-41x^{3}+18x^{2}-23+4x}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Utvid \left(x-2\right)^{2}\left(x+1\right)^{2}.

$$\frac{4x^{8}-8x^{7}-21x^{6}+19x^{4}+41x^{5}-41x^{3}+18x^{2}-23+4x}{x^{4}-2x^{3}-3x^{2}+4x+4}$$

Utvid

$$\frac{4x^{8}-8x^{7}-21x^{6}+41x^{5}+19x^{4}-41x^{3}+18x^{2}+4x-23}{\left(\left(x-2\right)\left(x+1\right)\right)^{2}}$$

Korte løsningstrinn

Hvis du vil legge til eller trekke fra uttrykk, kan du utvide dem for å gjøre nevnerne like. Multipliser 2x^{2} ganger \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}.

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

Siden \frac{2x^{2}\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} og \frac{1}{\left(x-2\right)\left(x+1\right)} har samme nevner, kan du subtrahere dem ved å subtrahere tellerne.

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

Utfør multiplikasjonene i 2x^{2}\left(x-2\right)\left(x+1\right)-1.

$$\left(\frac{2x^{4}+2x^{3}-4x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)}\right)^{2}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Kombiner like ledd i 2x^{4}+2x^{3}-4x^{3}-4x^{2}-1.

$$\left(\frac{2x^{4}-2x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)}\right)^{2}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Hvis du vil heve \frac{2x^{4}-2x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)} i en potens, øker du både telleren og nevneren i en potens, og deler deretter.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(\left(x-2\right)\left(x+1\right)\right)^{2}}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Utvid \left(\left(x-2\right)\left(x+1\right)\right)^{2}.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-8\left(2x^{2}-1\right)+7\left(x-1\right)\left(x+2\right)$$

Bruk den distributive lov til å multiplisere -8 med 2x^{2}-1.

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

Bruk den distributive lov til å multiplisere 7 med x-1.

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

Bruk den distributive lov til å multiplisere 7x-7 med x+2 og kombinere like ledd.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-16x^{2}+8+7x^{2}+7x-14$$

Kombiner -16x^{2} og 7x^{2} for å få -9x^{2}.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-9x^{2}+8+7x-14$$

Trekk fra 14 fra 8 for å få -6.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}-9x^{2}-6+7x$$

Hvis du vil legge til eller trekke fra uttrykk, kan du utvide dem for å gjøre nevnerne like. Multipliser -9x^{2}-6+7x ganger \frac{\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}+\frac{\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Siden \frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}} og \frac{\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}} har samme nevner, kan du legge dem sammen ved å legge sammen tellerne.

$$\frac{\left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}+\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Utfør multiplikasjonene i \left(2x^{4}-2x^{3}-4x^{2}-1\right)^{2}+\left(-9x^{2}-6+7x\right)\left(x-2\right)^{2}\left(x+1\right)^{2}.

$$\frac{4x^{8}-4x^{7}-8x^{6}-2x^{4}-4x^{7}+4x^{6}+8x^{5}+2x^{3}-8x^{6}+8x^{5}+16x^{4}+4x^{2}-2x^{4}+2x^{3}+4x^{2}+1-9x^{6}+18x^{5}+27x^{4}-36x^{3}-36x^{2}-6x^{4}+12x^{3}+18x^{2}-24x-24+7x^{5}-14x^{4}-21x^{3}+28x^{2}+28x}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Kombiner like ledd i 4x^{8}-4x^{7}-8x^{6}-2x^{4}-4x^{7}+4x^{6}+8x^{5}+2x^{3}-8x^{6}+8x^{5}+16x^{4}+4x^{2}-2x^{4}+2x^{3}+4x^{2}+1-9x^{6}+18x^{5}+27x^{4}-36x^{3}-36x^{2}-6x^{4}+12x^{3}+18x^{2}-24x-24+7x^{5}-14x^{4}-21x^{3}+28x^{2}+28x.

$$\frac{4x^{8}-8x^{7}-21x^{6}+19x^{4}+41x^{5}-41x^{3}+18x^{2}-23+4x}{\left(x-2\right)^{2}\left(x+1\right)^{2}}$$

Utvid \left(x-2\right)^{2}\left(x+1\right)^{2}.

$$\frac{4x^{8}-8x^{7}-21x^{6}+19x^{4}+41x^{5}-41x^{3}+18x^{2}-23+4x}{x^{4}-2x^{3}-3x^{2}+4x+4}$$