$$\frac{ x+y }{ { x }^{ 2 } -xy+ { y }^{ 2 } } - \frac{ 1 }{ x+y } + \frac{ 3 { x }^{ 2 } }{ { x }^{ 3 } + { y }^{ 3 } }$$
Evaluate
$\frac{3x}{x^{2}-xy+y^{2}}$
Short Solution Steps
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x^{2}-xy+y^{2}$ and $x+y$ is $\left(x+y\right)\left(x^{2}-xy+y^{2}\right)$. Multiply $\frac{x+y}{x^{2}-xy+y^{2}}$ times $\frac{x+y}{x+y}$. Multiply $\frac{1}{x+y}$ times $\frac{x^{2}-xy+y^{2}}{x^{2}-xy+y^{2}}$.
Since $\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ and $\frac{x^{2}-xy+y^{2}}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ have the same denominator, subtract them by subtracting their numerators.
Since $\frac{3xy}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ and $\frac{3x^{2}}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ have the same denominator, add them by adding their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x^{2}-xy+y^{2}$ and $x+y$ is $\left(x+y\right)\left(x^{2}-xy+y^{2}\right)$. Multiply $\frac{x+y}{x^{2}-xy+y^{2}}$ times $\frac{x+y}{x+y}$. Multiply $\frac{1}{x+y}$ times $\frac{x^{2}-xy+y^{2}}{x^{2}-xy+y^{2}}$.
Since $\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ and $\frac{x^{2}-xy+y^{2}}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ have the same denominator, subtract them by subtracting their numerators.
Since $\frac{3xy}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ and $\frac{3x^{2}}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$ have the same denominator, add them by adding their numerators.
