Solve (x^{4}-2x^2-6x-1)+2(x^3-6x^2+4)-(2x-2)^3

$$( x ^ { 4 } - 2 x ^ { 2 } - 6 x - 1 ) + 2 ( x ^ { 3 } - 6 x ^ { 2 } + 4 ) - ( 2 x - 2 ) ^ { 3 }$$

Evaluate

$$x^{4}-6x^{3}+10x^{2}-30x+15$$

Solution Steps

Use the distributive property to multiply 2 by x^{3}-6x^{2}+4.

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

Combine -2x^{2} and -12x^{2} to get -14x^{2}.

$$x^{4}-14x^{2}-6x-1+2x^{3}+8-\left(2x-2\right)^{3}$$

Add -1 and 8 to get 7.

$$x^{4}-14x^{2}-6x+7+2x^{3}-\left(2x-2\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(2x-2\right)^{3}.

$$x^{4}-14x^{2}-6x+7+2x^{3}-\left(8x^{3}-24x^{2}+24x-8\right)$$

To find the opposite of 8x^{3}-24x^{2}+24x-8, find the opposite of each term.

$$x^{4}-14x^{2}-6x+7+2x^{3}-8x^{3}+24x^{2}-24x+8$$

Combine 2x^{3} and -8x^{3} to get -6x^{3}.

$$x^{4}-14x^{2}-6x+7-6x^{3}+24x^{2}-24x+8$$

Combine -14x^{2} and 24x^{2} to get 10x^{2}.

$$x^{4}+10x^{2}-6x+7-6x^{3}-24x+8$$

Combine -6x and -24x to get -30x.

$$x^{4}+10x^{2}-30x+7-6x^{3}+8$$

Add 7 and 8 to get 15.

$$x^{4}+10x^{2}-30x+15-6x^{3}$$

Expand

$$x^{4}-6x^{3}+10x^{2}-30x+15$$

Solution Steps

Use the distributive property to multiply 2 by x^{3}-6x^{2}+4.

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

Combine -2x^{2} and -12x^{2} to get -14x^{2}.

$$x^{4}-14x^{2}-6x-1+2x^{3}+8-\left(2x-2\right)^{3}$$

Add -1 and 8 to get 7.

$$x^{4}-14x^{2}-6x+7+2x^{3}-\left(2x-2\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(2x-2\right)^{3}.

$$x^{4}-14x^{2}-6x+7+2x^{3}-\left(8x^{3}-24x^{2}+24x-8\right)$$

To find the opposite of 8x^{3}-24x^{2}+24x-8, find the opposite of each term.

$$x^{4}-14x^{2}-6x+7+2x^{3}-8x^{3}+24x^{2}-24x+8$$

Combine 2x^{3} and -8x^{3} to get -6x^{3}.

$$x^{4}-14x^{2}-6x+7-6x^{3}+24x^{2}-24x+8$$

Combine -14x^{2} and 24x^{2} to get 10x^{2}.

$$x^{4}+10x^{2}-6x+7-6x^{3}-24x+8$$

Combine -6x and -24x to get -30x.

$$x^{4}+10x^{2}-30x+7-6x^{3}+8$$

Add 7 and 8 to get 15.

$$x^{4}+10x^{2}-30x+15-6x^{3}$$