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

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

Evaluate

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

Solution Steps

Multiply x and x to get x^{2}.

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

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

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

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

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

Combine x^{2} and 3x^{2} to get 4x^{2}.

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

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

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

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

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

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

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

Combine -3x and x to get -2x.

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

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

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

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

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

Expand

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

Solution Steps

Multiply x and x to get x^{2}.

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

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

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

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

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

Combine x^{2} and 3x^{2} to get 4x^{2}.

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

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

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

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

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

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

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

Combine -3x and x to get -2x.

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

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

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

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

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