## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

The simplified form of the rational expression $\frac{2x}{{{x}^{2}}-3x}\div \left( x-3 \right)$ is$\frac{2}{{{\left( x-3 \right)}^{2}}}$.
$\frac{2x}{{{x}^{2}}-3x}\div \left( x-3 \right)$ $\frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C}$ The reciprocal of $\frac{\left( x-3 \right)}{1}$ is $\frac{1}{\left( x-3 \right)}$ So, multiply the reciprocal of the divisor, \begin{align} & \frac{2x}{{{x}^{2}}-3x}\div \left( x-3 \right)=\frac{2x}{{{x}^{2}}-3x}\cdot \frac{1}{\left( x-3 \right)} \\ & =\frac{\left( 2x \right)1}{\left( {{x}^{2}}-3x \right)\left( x-3 \right)} \end{align} $\left( {{x}^{2}}-3x \right)\left( x-3 \right)=x\left( x-3 \right)\left( x-3 \right)$ So, the rational expression becomes, $\frac{2x}{{{x}^{2}}-3x}\div \left( x-3 \right)=\frac{\left( 2x \right)}{x\left( x-3 \right)\left( x-3 \right)}$ Regroup and remove the factor equal to 1, \begin{align} & \frac{2x}{{{x}^{2}}-3x}\div \left( x-3 \right)=\frac{2x}{x\left( x-3 \right)\left( x-3 \right)} \\ & =\frac{x}{x}\cdot \frac{2}{\left( x-3 \right)\left( x-3 \right)} \\ & =1\cdot \frac{2}{{{\left( x-3 \right)}^{2}}} \\ & =\frac{2}{{{\left( x-3 \right)}^{2}}} \end{align}