Solve left(x/x+x^2-1/x-x^2right)*(x-x^3/2)

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x}{x+x^{2}}.

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

Cancel out x in both numerator and denominator.

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

Factor x-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x\left(-x+1\right) is x\left(x+1\right)\left(-x+1\right). Multiply \frac{1}{x+1} times \frac{x\left(-x+1\right)}{x\left(-x+1\right)}. Multiply \frac{1}{x\left(-x+1\right)} times \frac{x+1}{x+1}.

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

Since \frac{x\left(-x+1\right)}{x\left(x+1\right)\left(-x+1\right)} and \frac{x+1}{x\left(x+1\right)\left(-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(-x+1\right)-\left(x+1\right).

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

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

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

Multiply \frac{-x^{2}-1}{x\left(x+1\right)\left(-x+1\right)} times \frac{x-x^{3}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -1-x. Extract the negative sign in -1+x.

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

Cancel out x\left(x+1\right)\left(-x+1\right) in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x}{x+x^{2}}.

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

Cancel out x in both numerator and denominator.

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

Factor x-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x\left(-x+1\right) is x\left(x+1\right)\left(-x+1\right). Multiply \frac{1}{x+1} times \frac{x\left(-x+1\right)}{x\left(-x+1\right)}. Multiply \frac{1}{x\left(-x+1\right)} times \frac{x+1}{x+1}.

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

Since \frac{x\left(-x+1\right)}{x\left(x+1\right)\left(-x+1\right)} and \frac{x+1}{x\left(x+1\right)\left(-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(-x+1\right)-\left(x+1\right).

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

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

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

Multiply \frac{-x^{2}-1}{x\left(x+1\right)\left(-x+1\right)} times \frac{x-x^{3}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -1-x. Extract the negative sign in -1+x.

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

Cancel out x\left(x+1\right)\left(-x+1\right) in both numerator and denominator.

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

Expand the expression.

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