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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in xx-1.

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

Express \frac{x^{2}-1}{x}\left(x^{2}+1\right) as a single fraction.

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

Express \frac{\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x}}{\left(x+1\right)\left(x-x^{3}\right)} as a single fraction.

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

Factor the expressions that are not already factored.

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

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

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

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

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

Expand the expression.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in xx-1.

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

Express \frac{x^{2}-1}{x}\left(x^{2}+1\right) as a single fraction.

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

Express \frac{\frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x}}{\left(x+1\right)\left(x-x^{3}\right)} as a single fraction.

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

Factor the expressions that are not already factored.

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

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

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

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

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

Expand the expression.

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