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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Combine 3x and -x to get 2x.

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

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

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

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

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

Expand

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Combine 3x and -x to get 2x.

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

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

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

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

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