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

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

Evaluate

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

Short Solution Steps

Factor x^{3}-1. Factor 2x+2.

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

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

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

Since \frac{x\times 2\left(x+1\right)}{2\left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)} and \frac{\left(x-1\right)\left(x^{2}+x+1\right)}{2\left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\times 2\left(x+1\right)-\left(x-1\right)\left(x^{2}+x+1\right).

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

Combine like terms in 2x^{2}+2x-x^{3}-x^{2}-x+x^{2}+x+1.

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

Factor x^{2}-1.

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

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

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

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

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

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

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

Combine like terms in 2x^{2}+2x-x^{3}+1-2x^{2}-2x-2.

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Factor 2x^{3}-2.

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

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand 2\left(x^{2}+x+1\right).

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

Factor

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