Solve (x/x^2-2x+1-x+1/x^2-xdiv1/x-1

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

Evaluate

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

Short Solution Steps

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

Cancel out x-1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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