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

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

Evaluate

$$x^{3}$$

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}.

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

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

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

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

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

Combine -3x^{2} and 3x^{2} to get 0.

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

Combine 3x and -6x to get -3x.

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

Add -1 and 3 to get 2.

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

Use the distributive property to multiply 3 by x-1.

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

Combine -3x and 3x to get 0.

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

Subtract 3 from 2 to get -1.

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

Add -1 and 1 to get 0.

$$x^{3}$$

Expand

$$x^{3}$$

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}.

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

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

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

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

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

Combine -3x^{2} and 3x^{2} to get 0.

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

Combine 3x and -6x to get -3x.

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

Add -1 and 3 to get 2.

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

Use the distributive property to multiply 3 by x-1.

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

Combine -3x and 3x to get 0.

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

Subtract 3 from 2 to get -1.

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

Add -1 and 1 to get 0.

$$x^{3}$$