Solve -(x-1)^3-2(x-1)^2+5(x-1)+6

$$- ( x - 1 ) ^ { 3 } - 2 ( x - 1 ) ^ { 2 } + 5 ( x - 1 ) + 6$$

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-1\right)^{3}.

$$-\left(x^{3}-3x^{2}+3x-1\right)-2\left(x-1\right)^{2}+5\left(x-1\right)+6$$

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

$$-x^{3}+3x^{2}-3x+1-2\left(x-1\right)^{2}+5\left(x-1\right)+6$$

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

$$-x^{3}+3x^{2}-3x+1-2\left(x^{2}-2x+1\right)+5\left(x-1\right)+6$$

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

$$-x^{3}+3x^{2}-3x+1-2x^{2}+4x-2+5\left(x-1\right)+6$$

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

$$-x^{3}+x^{2}-3x+1+4x-2+5\left(x-1\right)+6$$

Combine -3x and 4x to get x.

$$-x^{3}+x^{2}+x+1-2+5\left(x-1\right)+6$$

Subtract 2 from 1 to get -1.

$$-x^{3}+x^{2}+x-1+5\left(x-1\right)+6$$

Use the distributive property to multiply 5 by x-1.

$$-x^{3}+x^{2}+x-1+5x-5+6$$

Combine x and 5x to get 6x.

$$-x^{3}+x^{2}+6x-1-5+6$$

Subtract 5 from -1 to get -6.

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

Add -6 and 6 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-1\right)^{3}.

$$-\left(x^{3}-3x^{2}+3x-1\right)-2\left(x-1\right)^{2}+5\left(x-1\right)+6$$

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

$$-x^{3}+3x^{2}-3x+1-2\left(x-1\right)^{2}+5\left(x-1\right)+6$$

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

$$-x^{3}+3x^{2}-3x+1-2\left(x^{2}-2x+1\right)+5\left(x-1\right)+6$$

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

$$-x^{3}+3x^{2}-3x+1-2x^{2}+4x-2+5\left(x-1\right)+6$$

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

$$-x^{3}+x^{2}-3x+1+4x-2+5\left(x-1\right)+6$$

Combine -3x and 4x to get x.

$$-x^{3}+x^{2}+x+1-2+5\left(x-1\right)+6$$

Subtract 2 from 1 to get -1.

$$-x^{3}+x^{2}+x-1+5\left(x-1\right)+6$$

Use the distributive property to multiply 5 by x-1.

$$-x^{3}+x^{2}+x-1+5x-5+6$$

Combine x and 5x to get 6x.

$$-x^{3}+x^{2}+6x-1-5+6$$

Subtract 5 from -1 to get -6.

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

Add -6 and 6 to get 0.

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