Solve (x+1/x^2-x-x/x^2-2x+1)div1/x+x^2/(x-1)^2

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Divide \frac{-1}{x\left(x-1\right)^{2}} by \frac{1}{x} by multiplying \frac{-1}{x\left(x-1\right)^{2}} by the reciprocal of \frac{1}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Divide \frac{-1}{x\left(x-1\right)^{2}} by \frac{1}{x} by multiplying \frac{-1}{x\left(x-1\right)^{2}} by the reciprocal of \frac{1}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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