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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x+1-2x+2.

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -1-x.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

To find the opposite of x-2, find the opposite of each term.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x+1-2x+2.

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -1-x.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

To find the opposite of x-2, find the opposite of each term.

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