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

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

Evaluate

$$x^{3}-5x^{2}+8x-a-4$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

$$x^{3}-5x^{2}+3x-1+4x-2+x-1-a$$

Combine 3x and 4x to get 7x.

$$x^{3}-5x^{2}+7x-1-2+x-1-a$$

Subtract 2 from -1 to get -3.

$$x^{3}-5x^{2}+7x-3+x-1-a$$

Combine 7x and x to get 8x.

$$x^{3}-5x^{2}+8x-3-1-a$$

Subtract 1 from -3 to get -4.

$$x^{3}-5x^{2}+8x-4-a$$

Expand

$$x^{3}-5x^{2}+8x-a-4$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

$$x^{3}-5x^{2}+3x-1+4x-2+x-1-a$$

Combine 3x and 4x to get 7x.

$$x^{3}-5x^{2}+7x-1-2+x-1-a$$

Subtract 2 from -1 to get -3.

$$x^{3}-5x^{2}+7x-3+x-1-a$$

Combine 7x and x to get 8x.

$$x^{3}-5x^{2}+8x-3-1-a$$

Subtract 1 from -3 to get -4.

$$x^{3}-5x^{2}+8x-4-a$$