Solve 1/x+x^2+1/x-x^2+x+3/1-x^2

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

Evaluate

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

Short Solution Steps

Factor x+x^{2}. Factor x-x^{2}.

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

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

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

Since \frac{-x+1}{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, add them by adding their numerators.

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

Combine like terms in -x+1+x+1.

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

Factor 1-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right)\left(-x+1\right) and \left(x-1\right)\left(-x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{2}{x\left(x+1\right)\left(-x+1\right)} times \frac{-1}{-1}. Multiply \frac{x+3}{\left(x-1\right)\left(-x-1\right)} times \frac{-x}{-x}.

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

Since \frac{2\left(-1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{\left(x+3\right)\left(-1\right)x}{x\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(-1\right)+\left(x+3\right)\left(-1\right)x.

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

Factor the expressions that are not already factored in \frac{-2-x^{2}-3x}{x\left(x-1\right)\left(x+1\right)}.

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

Extract the negative sign in -1-x.

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

Cancel out x+1 in both numerator and denominator.

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

Expand x\left(x-1\right).

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

To find the opposite of x+2, find the opposite of each term.

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

Expand

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

Short Solution Steps

Factor x+x^{2}. Factor x-x^{2}.

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

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

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

Since \frac{-x+1}{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, add them by adding their numerators.

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

Combine like terms in -x+1+x+1.

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

Factor 1-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right)\left(-x+1\right) and \left(x-1\right)\left(-x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{2}{x\left(x+1\right)\left(-x+1\right)} times \frac{-1}{-1}. Multiply \frac{x+3}{\left(x-1\right)\left(-x-1\right)} times \frac{-x}{-x}.

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

Since \frac{2\left(-1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{\left(x+3\right)\left(-1\right)x}{x\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(-1\right)+\left(x+3\right)\left(-1\right)x.

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

Factor the expressions that are not already factored in \frac{-2-x^{2}-3x}{x\left(x-1\right)\left(x+1\right)}.

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

Extract the negative sign in -1-x.

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

Cancel out x+1 in both numerator and denominator.

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

Expand x\left(x-1\right).

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

To find the opposite of x+2, find the opposite of each term.

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