Solve 4-y/9-y^2+1-y/9+6y+y^2-2+y/y^2+6y+9

$$\frac { 4 - y } { 9 - y ^ { 2 } } + \frac { 1 - y } { 9 + 6 y + y ^ { 2 } } - \frac { 2 + y } { y ^ { 2 } + 6 y + 9 }$$

Evaluate

$$\frac{y^{2}-4y+9}{\left(3-y\right)\left(y+3\right)^{2}}$$

Short Solution Steps

Factor 9-y^{2}. Factor 9+6y+y^{2}.

$$\frac{4-y}{\left(y-3\right)\left(-y-3\right)}+\frac{1-y}{\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-3\right)\left(-y-3\right) and \left(y+3\right)^{2} is \left(y-3\right)\left(y+3\right)^{2}. Multiply \frac{4-y}{\left(y-3\right)\left(-y-3\right)} times \frac{-\left(y+3\right)}{-\left(y+3\right)}. Multiply \frac{1-y}{\left(y+3\right)^{2}} times \frac{y-3}{y-3}.

$$\frac{\left(4-y\right)\left(-1\right)\left(y+3\right)}{\left(y-3\right)\left(y+3\right)^{2}}+\frac{\left(1-y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Since \frac{\left(4-y\right)\left(-1\right)\left(y+3\right)}{\left(y-3\right)\left(y+3\right)^{2}} and \frac{\left(1-y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4-y\right)\left(-1\right)\left(y+3\right)+\left(1-y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Do the multiplications in \left(4-y\right)\left(-1\right)\left(y+3\right)+\left(1-y\right)\left(y-3\right).

$$\frac{-4y-12+y^{2}+3y+y-3-y^{2}+3y}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Combine like terms in -4y-12+y^{2}+3y+y-3-y^{2}+3y.

$$\frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Factor y^{2}+6y+9.

$$\frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{\left(y+3\right)^{2}}$$

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

$$\frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{\left(2+y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}$$

Since \frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}} and \frac{\left(2+y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3y-15-\left(2+y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}$$

Do the multiplications in 3y-15-\left(2+y\right)\left(y-3\right).

$$\frac{3y-15-2y+6-y^{2}+3y}{\left(y-3\right)\left(y+3\right)^{2}}$$

Combine like terms in 3y-15-2y+6-y^{2}+3y.

$$\frac{4y-9-y^{2}}{\left(y-3\right)\left(y+3\right)^{2}}$$

Expand \left(y-3\right)\left(y+3\right)^{2}.

$$\frac{4y-9-y^{2}}{y^{3}+3y^{2}-9y-27}$$

Expand

$$\frac{y^{2}-4y+9}{\left(3-y\right)\left(y+3\right)^{2}}$$

Short Solution Steps

Factor 9-y^{2}. Factor 9+6y+y^{2}.

$$\frac{4-y}{\left(y-3\right)\left(-y-3\right)}+\frac{1-y}{\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-3\right)\left(-y-3\right) and \left(y+3\right)^{2} is \left(y-3\right)\left(y+3\right)^{2}. Multiply \frac{4-y}{\left(y-3\right)\left(-y-3\right)} times \frac{-\left(y+3\right)}{-\left(y+3\right)}. Multiply \frac{1-y}{\left(y+3\right)^{2}} times \frac{y-3}{y-3}.

$$\frac{\left(4-y\right)\left(-1\right)\left(y+3\right)}{\left(y-3\right)\left(y+3\right)^{2}}+\frac{\left(1-y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Since \frac{\left(4-y\right)\left(-1\right)\left(y+3\right)}{\left(y-3\right)\left(y+3\right)^{2}} and \frac{\left(1-y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4-y\right)\left(-1\right)\left(y+3\right)+\left(1-y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Do the multiplications in \left(4-y\right)\left(-1\right)\left(y+3\right)+\left(1-y\right)\left(y-3\right).

$$\frac{-4y-12+y^{2}+3y+y-3-y^{2}+3y}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Combine like terms in -4y-12+y^{2}+3y+y-3-y^{2}+3y.

$$\frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{y^{2}+6y+9}$$

Factor y^{2}+6y+9.

$$\frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{2+y}{\left(y+3\right)^{2}}$$

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

$$\frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}}-\frac{\left(2+y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}$$

Since \frac{3y-15}{\left(y-3\right)\left(y+3\right)^{2}} and \frac{\left(2+y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3y-15-\left(2+y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}}$$

Do the multiplications in 3y-15-\left(2+y\right)\left(y-3\right).

$$\frac{3y-15-2y+6-y^{2}+3y}{\left(y-3\right)\left(y+3\right)^{2}}$$

Combine like terms in 3y-15-2y+6-y^{2}+3y.

$$\frac{4y-9-y^{2}}{\left(y-3\right)\left(y+3\right)^{2}}$$

Expand \left(y-3\right)\left(y+3\right)^{2}.

$$\frac{4y-9-y^{2}}{y^{3}+3y^{2}-9y-27}$$