Solve (a-2)^{3}-7(a^3-1)-2(a^2+a-1)(a^2-a-1)+2a^4

$$( a - 2 ) ^ { 3 } - 7 ( a ^ { 3 } - 1 ) - 2 ( a ^ { 2 } + a - 1 ) ( a ^ { 2 } - a - 1 ) + 2 a ^ { 4 }$$

Evaluate

$$-6a^{3}+12a-3$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(a-2\right)^{3}.

$$a^{3}-6a^{2}+12a-8-7\left(a^{3}-1\right)-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Use the distributive property to multiply -7 by a^{3}-1.

$$a^{3}-6a^{2}+12a-8-7a^{3}+7-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Combine a^{3} and -7a^{3} to get -6a^{3}.

$$-6a^{3}-6a^{2}+12a-8+7-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Add -8 and 7 to get -1.

$$-6a^{3}-6a^{2}+12a-1-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Use the distributive property to multiply -2 by a^{2}+a-1.

$$-6a^{3}-6a^{2}+12a-1+\left(-2a^{2}-2a+2\right)\left(a^{2}-a-1\right)+2a^{4}$$

Use the distributive property to multiply -2a^{2}-2a+2 by a^{2}-a-1 and combine like terms.

$$-6a^{3}-6a^{2}+12a-1-2a^{4}+6a^{2}-2+2a^{4}$$

Combine -6a^{2} and 6a^{2} to get 0.

$$-6a^{3}+12a-1-2a^{4}-2+2a^{4}$$

Subtract 2 from -1 to get -3.

$$-6a^{3}+12a-3-2a^{4}+2a^{4}$$

Combine -2a^{4} and 2a^{4} to get 0.

$$-6a^{3}+12a-3$$

Expand

$$-6a^{3}+12a-3$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(a-2\right)^{3}.

$$a^{3}-6a^{2}+12a-8-7\left(a^{3}-1\right)-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Use the distributive property to multiply -7 by a^{3}-1.

$$a^{3}-6a^{2}+12a-8-7a^{3}+7-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Combine a^{3} and -7a^{3} to get -6a^{3}.

$$-6a^{3}-6a^{2}+12a-8+7-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Add -8 and 7 to get -1.

$$-6a^{3}-6a^{2}+12a-1-2\left(a^{2}+a-1\right)\left(a^{2}-a-1\right)+2a^{4}$$

Use the distributive property to multiply -2 by a^{2}+a-1.

$$-6a^{3}-6a^{2}+12a-1+\left(-2a^{2}-2a+2\right)\left(a^{2}-a-1\right)+2a^{4}$$

Use the distributive property to multiply -2a^{2}-2a+2 by a^{2}-a-1 and combine like terms.

$$-6a^{3}-6a^{2}+12a-1-2a^{4}+6a^{2}-2+2a^{4}$$

Combine -6a^{2} and 6a^{2} to get 0.

$$-6a^{3}+12a-1-2a^{4}-2+2a^{4}$$

Subtract 2 from -1 to get -3.

$$-6a^{3}+12a-3-2a^{4}+2a^{4}$$

Combine -2a^{4} and 2a^{4} to get 0.

$$-6a^{3}+12a-3$$