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

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

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}.

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

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.

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

Add -1 and 2 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.

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

Add 1 and 5 to get 6.

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

Subtract 6 from 6 to get 0.

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

Expand

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

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}.

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

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.

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

Add -1 and 2 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.

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

Add 1 and 5 to get 6.

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

Subtract 6 from 6 to get 0.

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