Solve (x^3-2x^2+3x-4)(x-1)-(2x-3)(x^2-x+1)

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

Evaluate

$$x^{4}-5x^{3}+10x^{2}-12x+7$$

Solution Steps

Use the distributive property to multiply x^{3}-2x^{2}+3x-4 by x-1 and combine like terms.

$$x^{4}-3x^{3}+5x^{2}-7x+4-\left(2x-3\right)\left(x^{2}-x+1\right)$$

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

$$x^{4}-3x^{3}+5x^{2}-7x+4-\left(2x^{3}-5x^{2}+5x-3\right)$$

To find the opposite of 2x^{3}-5x^{2}+5x-3, find the opposite of each term.

$$x^{4}-3x^{3}+5x^{2}-7x+4-2x^{3}+5x^{2}-5x+3$$

Combine -3x^{3} and -2x^{3} to get -5x^{3}.

$$x^{4}-5x^{3}+5x^{2}-7x+4+5x^{2}-5x+3$$

Combine 5x^{2} and 5x^{2} to get 10x^{2}.

$$x^{4}-5x^{3}+10x^{2}-7x+4-5x+3$$

Combine -7x and -5x to get -12x.

$$x^{4}-5x^{3}+10x^{2}-12x+4+3$$

Add 4 and 3 to get 7.

$$x^{4}-5x^{3}+10x^{2}-12x+7$$

Expand

$$x^{4}-5x^{3}+10x^{2}-12x+7$$

Solution Steps

Use the distributive property to multiply x^{3}-2x^{2}+3x-4 by x-1 and combine like terms.

$$x^{4}-3x^{3}+5x^{2}-7x+4-\left(2x-3\right)\left(x^{2}-x+1\right)$$

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

$$x^{4}-3x^{3}+5x^{2}-7x+4-\left(2x^{3}-5x^{2}+5x-3\right)$$

To find the opposite of 2x^{3}-5x^{2}+5x-3, find the opposite of each term.

$$x^{4}-3x^{3}+5x^{2}-7x+4-2x^{3}+5x^{2}-5x+3$$

Combine -3x^{3} and -2x^{3} to get -5x^{3}.

$$x^{4}-5x^{3}+5x^{2}-7x+4+5x^{2}-5x+3$$

Combine 5x^{2} and 5x^{2} to get 10x^{2}.

$$x^{4}-5x^{3}+10x^{2}-7x+4-5x+3$$

Combine -7x and -5x to get -12x.

$$x^{4}-5x^{3}+10x^{2}-12x+4+3$$

Add 4 and 3 to get 7.

$$x^{4}-5x^{3}+10x^{2}-12x+7$$