Solve 2/x(x-1)-5/x^2(x-1)+3/(x-1)(x+1)

$$\frac { 2 } { x ( x - 1 ) } - \frac { 5 } { x ^ { 2 } ( x - 1 ) } + \frac { 3 } { ( x - 1 ) ( x + 1 ) }$$

Evaluate

$$\frac{5x^{2}-3x-5}{x^{2}\left(x^{2}-1\right)}$$

Short Solution Steps

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

$$\frac{2x}{\left(x-1\right)x^{2}}-\frac{5}{\left(x-1\right)x^{2}}+\frac{3}{\left(x-1\right)\left(x+1\right)}$$

Since \frac{2x}{\left(x-1\right)x^{2}} and \frac{5}{\left(x-1\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x-5}{\left(x-1\right)x^{2}}+\frac{3}{\left(x-1\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{2} and \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right)x^{2}. Multiply \frac{2x-5}{\left(x-1\right)x^{2}} times \frac{x+1}{x+1}. Multiply \frac{3}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}}{x^{2}}.

$$\frac{\left(2x-5\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{2}}+\frac{3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}}$$

Since \frac{\left(2x-5\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{2}} and \frac{3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x-5\right)\left(x+1\right)+3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}}$$

Do the multiplications in \left(2x-5\right)\left(x+1\right)+3x^{2}.

$$\frac{2x^{2}+2x-5x-5+3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}}$$

Combine like terms in 2x^{2}+2x-5x-5+3x^{2}.

$$\frac{5x^{2}-3x-5}{\left(x-1\right)\left(x+1\right)x^{2}}$$

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

$$\frac{5x^{2}-3x-5}{x^{4}-x^{2}}$$

Expand

$$\frac{5x^{2}-3x-5}{x^{2}\left(x^{2}-1\right)}$$

Short Solution Steps

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

$$\frac{2x}{\left(x-1\right)x^{2}}-\frac{5}{\left(x-1\right)x^{2}}+\frac{3}{\left(x-1\right)\left(x+1\right)}$$

Since \frac{2x}{\left(x-1\right)x^{2}} and \frac{5}{\left(x-1\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x-5}{\left(x-1\right)x^{2}}+\frac{3}{\left(x-1\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{2} and \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right)x^{2}. Multiply \frac{2x-5}{\left(x-1\right)x^{2}} times \frac{x+1}{x+1}. Multiply \frac{3}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}}{x^{2}}.

$$\frac{\left(2x-5\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{2}}+\frac{3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}}$$

Since \frac{\left(2x-5\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{2}} and \frac{3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x-5\right)\left(x+1\right)+3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}}$$

Do the multiplications in \left(2x-5\right)\left(x+1\right)+3x^{2}.

$$\frac{2x^{2}+2x-5x-5+3x^{2}}{\left(x-1\right)\left(x+1\right)x^{2}}$$

Combine like terms in 2x^{2}+2x-5x-5+3x^{2}.

$$\frac{5x^{2}-3x-5}{\left(x-1\right)\left(x+1\right)x^{2}}$$

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

$$\frac{5x^{2}-3x-5}{x^{4}-x^{2}}$$