Solve (x-1)(x^3-2x^2+1)-(x+1)(x^3+2x^2+1)

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

Evaluate

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

Solution Steps

Use the distributive property to multiply x-1 by x^{3}-2x^{2}+1 and combine like terms.

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

Use the distributive property to multiply x+1 by x^{3}+2x^{2}+1 and combine like terms.

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

To find the opposite of x^{4}+3x^{3}+x+2x^{2}+1, find the opposite of each term.

$$x^{4}-3x^{3}+x+2x^{2}-1-x^{4}-3x^{3}-x-2x^{2}-1$$

Combine x^{4} and -x^{4} to get 0.

$$-3x^{3}+x+2x^{2}-1-3x^{3}-x-2x^{2}-1$$

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

$$-6x^{3}+x+2x^{2}-1-x-2x^{2}-1$$

Combine x and -x to get 0.

$$-6x^{3}+2x^{2}-1-2x^{2}-1$$

Combine 2x^{2} and -2x^{2} to get 0.

$$-6x^{3}-1-1$$

Subtract 1 from -1 to get -2.

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

Expand

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

Solution Steps

Use the distributive property to multiply x-1 by x^{3}-2x^{2}+1 and combine like terms.

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

Use the distributive property to multiply x+1 by x^{3}+2x^{2}+1 and combine like terms.

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

To find the opposite of x^{4}+3x^{3}+x+2x^{2}+1, find the opposite of each term.

$$x^{4}-3x^{3}+x+2x^{2}-1-x^{4}-3x^{3}-x-2x^{2}-1$$

Combine x^{4} and -x^{4} to get 0.

$$-3x^{3}+x+2x^{2}-1-3x^{3}-x-2x^{2}-1$$

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

$$-6x^{3}+x+2x^{2}-1-x-2x^{2}-1$$

Combine x and -x to get 0.

$$-6x^{3}+2x^{2}-1-2x^{2}-1$$

Combine 2x^{2} and -2x^{2} to get 0.

$$-6x^{3}-1-1$$

Subtract 1 from -1 to get -2.

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