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

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

Evaluate

$$\frac{x}{x+3}$$

Short Solution Steps

Factor x^{3}-1.

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

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 x-1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{1}{x-1} times \frac{x^{2}+x+1}{x^{2}+x+1}.

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

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

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

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

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

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

Divide \frac{x}{x^{2}+x+1} by \frac{x+3}{x^{2}+x+1} by multiplying \frac{x}{x^{2}+x+1} by the reciprocal of \frac{x+3}{x^{2}+x+1}.

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

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

$$\frac{x}{x+3}$$

Expand

$$\frac{x}{x+3}$$

Short Solution Steps

Factor x^{3}-1.

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

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 x-1 is \left(x-1\right)\left(x^{2}+x+1\right). Multiply \frac{1}{x-1} times \frac{x^{2}+x+1}{x^{2}+x+1}.

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

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

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

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

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

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

Divide \frac{x}{x^{2}+x+1} by \frac{x+3}{x^{2}+x+1} by multiplying \frac{x}{x^{2}+x+1} by the reciprocal of \frac{x+3}{x^{2}+x+1}.

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

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

$$\frac{x}{x+3}$$