Solve (1-1/2)(1+x+x^2/1-x)*(1-2x^2/1+x^2)

$$( 1 - \frac { 1 } { 2 } ) ( 1 + \frac { x + x ^ { 2 } } { 1 - x } ) \cdot ( 1 - \frac { 2 x ^ { 2 } } { 1 + x ^ { 2 } } )$$

Evaluate

$$\frac{x+1}{2}$$

Short Solution Steps

Subtract \frac{1}{2} from 1 to get \frac{1}{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1-x}{1-x}.

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

Since \frac{1-x}{1-x} and \frac{x+x^{2}}{1-x} have the same denominator, add them by adding their numerators.

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

Combine like terms in 1-x+x+x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1+x^{2}}{1+x^{2}}.

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

Since \frac{1+x^{2}}{1+x^{2}} and \frac{2x^{2}}{1+x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2}\times \frac{1+x^{2}}{1-x}\times \frac{1+x^{2}-2x^{2}}{1+x^{2}}$$

Combine like terms in 1+x^{2}-2x^{2}.

$$\frac{1}{2}\times \frac{1+x^{2}}{1-x}\times \frac{1-x^{2}}{1+x^{2}}$$

Multiply \frac{1}{2} times \frac{1+x^{2}}{1-x} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{1+x^{2}}{2\left(1-x\right)} times \frac{1-x^{2}}{1+x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x^{2}+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -1+x.

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

Cancel out -x+1 in both numerator and denominator.

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

Expand the expression.

$$\frac{x+1}{2}$$

Expand

$$\frac{x+1}{2}$$

Short Solution Steps

Subtract \frac{1}{2} from 1 to get \frac{1}{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1-x}{1-x}.

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

Since \frac{1-x}{1-x} and \frac{x+x^{2}}{1-x} have the same denominator, add them by adding their numerators.

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

Combine like terms in 1-x+x+x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1+x^{2}}{1+x^{2}}.

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

Since \frac{1+x^{2}}{1+x^{2}} and \frac{2x^{2}}{1+x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2}\times \frac{1+x^{2}}{1-x}\times \frac{1+x^{2}-2x^{2}}{1+x^{2}}$$

Combine like terms in 1+x^{2}-2x^{2}.

$$\frac{1}{2}\times \frac{1+x^{2}}{1-x}\times \frac{1-x^{2}}{1+x^{2}}$$

Multiply \frac{1}{2} times \frac{1+x^{2}}{1-x} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{1+x^{2}}{2\left(1-x\right)} times \frac{1-x^{2}}{1+x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x^{2}+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -1+x.

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

Cancel out -x+1 in both numerator and denominator.

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

Expand the expression.

$$\frac{x+1}{2}$$