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

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

Evaluate

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

Short Solution Steps

Cancel out x+1 in both numerator and denominator.

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

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Cancel out x-1 in both numerator and denominator.

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

Multiply \frac{x^{2}}{x^{2}-1} times \frac{x-1}{x} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x in both numerator and denominator.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

Expand \left(x-1\right)\left(x+1\right).

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

Expand

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

Short Solution Steps

Cancel out x+1 in both numerator and denominator.

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

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Cancel out x-1 in both numerator and denominator.

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

Multiply \frac{x^{2}}{x^{2}-1} times \frac{x-1}{x} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x in both numerator and denominator.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

Expand \left(x-1\right)\left(x+1\right).

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