Solve -(x-2)^3-5(x-2)^2-2(x-2)+8

$$- ( x - 2 ) ^ { 3 } - 5 ( x - 2 ) ^ { 2 } - 2 ( x - 2 ) + 8$$

Evaluate

$$-x\left(x-3\right)\left(x+2\right)$$

Solution Steps

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

$$-\left(x^{3}-6x^{2}+12x-8\right)-5\left(x-2\right)^{2}-2\left(x-2\right)+8$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$-x^{3}+6x^{2}-12x+8-5\left(x-2\right)^{2}-2\left(x-2\right)+8$$

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

$$-x^{3}+6x^{2}-12x+8-5\left(x^{2}-4x+4\right)-2\left(x-2\right)+8$$

Use the distributive property to multiply -5 by x^{2}-4x+4.

$$-x^{3}+6x^{2}-12x+8-5x^{2}+20x-20-2\left(x-2\right)+8$$

Combine 6x^{2} and -5x^{2} to get x^{2}.

$$-x^{3}+x^{2}-12x+8+20x-20-2\left(x-2\right)+8$$

Combine -12x and 20x to get 8x.

$$-x^{3}+x^{2}+8x+8-20-2\left(x-2\right)+8$$

Subtract 20 from 8 to get -12.

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

Use the distributive property to multiply -2 by x-2.

$$-x^{3}+x^{2}+8x-12-2x+4+8$$

Combine 8x and -2x to get 6x.

$$-x^{3}+x^{2}+6x-12+4+8$$

Add -12 and 4 to get -8.

$$-x^{3}+x^{2}+6x-8+8$$

Add -8 and 8 to get 0.

$$-x^{3}+x^{2}+6x$$

Expand

$$6x+x^{2}-x^{3}$$

Solution Steps

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

$$-\left(x^{3}-6x^{2}+12x-8\right)-5\left(x-2\right)^{2}-2\left(x-2\right)+8$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$-x^{3}+6x^{2}-12x+8-5\left(x-2\right)^{2}-2\left(x-2\right)+8$$

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

$$-x^{3}+6x^{2}-12x+8-5\left(x^{2}-4x+4\right)-2\left(x-2\right)+8$$

Use the distributive property to multiply -5 by x^{2}-4x+4.

$$-x^{3}+6x^{2}-12x+8-5x^{2}+20x-20-2\left(x-2\right)+8$$

Combine 6x^{2} and -5x^{2} to get x^{2}.

$$-x^{3}+x^{2}-12x+8+20x-20-2\left(x-2\right)+8$$

Combine -12x and 20x to get 8x.

$$-x^{3}+x^{2}+8x+8-20-2\left(x-2\right)+8$$

Subtract 20 from 8 to get -12.

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

Use the distributive property to multiply -2 by x-2.

$$-x^{3}+x^{2}+8x-12-2x+4+8$$

Combine 8x and -2x to get 6x.

$$-x^{3}+x^{2}+6x-12+4+8$$

Add -12 and 4 to get -8.

$$-x^{3}+x^{2}+6x-8+8$$

Add -8 and 8 to get 0.

$$-x^{3}+x^{2}+6x$$