Solve y^2+2y+1-1/y^2-2y+1/frac{y{y-1}+y}

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

Evaluate

$$\frac{y^{2}-2}{y-1}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{2}+2y+1 times \frac{\left(y-1\right)^{2}}{\left(y-1\right)^{2}}.

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

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

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

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

$$\frac{\frac{y^{4}-2y^{3}+y^{2}+2y^{3}-4y^{2}+2y+y^{2}-2y+1-1}{\left(y-1\right)^{2}}}{\frac{y}{y-1}+y}$$

Combine like terms in y^{4}-2y^{3}+y^{2}+2y^{3}-4y^{2}+2y+y^{2}-2y+1-1.

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y}{y-1}+y}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{y-1}{y-1}.

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y}{y-1}+\frac{y\left(y-1\right)}{y-1}}$$

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

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y+y\left(y-1\right)}{y-1}}$$

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

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y+y^{2}-y}{y-1}}$$

Combine like terms in y+y^{2}-y.

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y^{2}}{y-1}}$$

Divide \frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}} by \frac{y^{2}}{y-1} by multiplying \frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}} by the reciprocal of \frac{y^{2}}{y-1}.

$$\frac{\left(y^{4}-2y^{2}\right)\left(y-1\right)}{\left(y-1\right)^{2}y^{2}}$$

Cancel out y-1 in both numerator and denominator.

$$\frac{y^{4}-2y^{2}}{\left(y-1\right)y^{2}}$$

Factor the expressions that are not already factored.

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

Cancel out y^{2} in both numerator and denominator.

$$\frac{y^{2}-2}{y-1}$$

Expand

$$\frac{y^{2}-2}{y-1}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{2}+2y+1 times \frac{\left(y-1\right)^{2}}{\left(y-1\right)^{2}}.

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

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

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

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

$$\frac{\frac{y^{4}-2y^{3}+y^{2}+2y^{3}-4y^{2}+2y+y^{2}-2y+1-1}{\left(y-1\right)^{2}}}{\frac{y}{y-1}+y}$$

Combine like terms in y^{4}-2y^{3}+y^{2}+2y^{3}-4y^{2}+2y+y^{2}-2y+1-1.

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y}{y-1}+y}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{y-1}{y-1}.

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y}{y-1}+\frac{y\left(y-1\right)}{y-1}}$$

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

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y+y\left(y-1\right)}{y-1}}$$

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

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y+y^{2}-y}{y-1}}$$

Combine like terms in y+y^{2}-y.

$$\frac{\frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}}}{\frac{y^{2}}{y-1}}$$

Divide \frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}} by \frac{y^{2}}{y-1} by multiplying \frac{y^{4}-2y^{2}}{\left(y-1\right)^{2}} by the reciprocal of \frac{y^{2}}{y-1}.

$$\frac{\left(y^{4}-2y^{2}\right)\left(y-1\right)}{\left(y-1\right)^{2}y^{2}}$$

Cancel out y-1 in both numerator and denominator.

$$\frac{y^{4}-2y^{2}}{\left(y-1\right)y^{2}}$$

Factor the expressions that are not already factored.

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

Cancel out y^{2} in both numerator and denominator.

$$\frac{y^{2}-2}{y-1}$$