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 løsningstrin

For tilføje eller fratrække udtryk skal du udvide dem for at gøre nævneren ens. Multiplicer 2x^{2} gange \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}.

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

Eftersom \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 den samme fællesnævner, kan du trække dem fra dem ved at trække deres tællere fra.

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

Lav multiplikationerne 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 ens led i 2x^{4}+2x^{3}-4x^{3}-4x^{2}-1.

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

For at hæve \frac{2x^{4}-2x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)} i en potens skal både tælleren og nævneren hæves i potensen og så divideres.

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

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

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

Brug fordelingsegenskaben til at multiplicere -8 med 2x^{2}-1.

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

Brug fordelingsegenskaben til at multiplicere 7 med x-1.

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

Brug fordelingsegenskaben til at multiplicere 7x-7 med x+2, og kombiner ens led.

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

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

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

Subtraher 14 fra 8 for at få -6.

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

For tilføje eller fratrække udtryk skal du udvide dem for at gøre nævneren ens. Multiplicer -9x^{2}-6+7x gange \frac{\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}.

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

Da \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 den samme fællesnævner, skal du addere dem ved at tilføje deres tællere.

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

Lav multiplikationerne 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 ens led 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.

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

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

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

Udvid

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

Korte løsningstrin

For tilføje eller fratrække udtryk skal du udvide dem for at gøre nævneren ens. Multiplicer 2x^{2} gange \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}.

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

Eftersom \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 den samme fællesnævner, kan du trække dem fra dem ved at trække deres tællere fra.

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

Lav multiplikationerne 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 ens led i 2x^{4}+2x^{3}-4x^{3}-4x^{2}-1.

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

For at hæve \frac{2x^{4}-2x^{3}-4x^{2}-1}{\left(x-2\right)\left(x+1\right)} i en potens skal både tælleren og nævneren hæves i potensen og så divideres.

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

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

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

Brug fordelingsegenskaben til at multiplicere -8 med 2x^{2}-1.

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

Brug fordelingsegenskaben til at multiplicere 7 med x-1.

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

Brug fordelingsegenskaben til at multiplicere 7x-7 med x+2, og kombiner ens led.

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

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

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

Subtraher 14 fra 8 for at få -6.

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

For tilføje eller fratrække udtryk skal du udvide dem for at gøre nævneren ens. Multiplicer -9x^{2}-6+7x gange \frac{\left(x-2\right)^{2}\left(x+1\right)^{2}}{\left(x-2\right)^{2}\left(x+1\right)^{2}}.

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

Da \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 den samme fællesnævner, skal du addere dem ved at tilføje deres tællere.

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

Lav multiplikationerne 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 ens led 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.

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

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

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