Question

For the following exercises, find $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$where $f\left(x\right)\mathrm{and}g\mathit{\left(}x\mathit{\right)}$are given.

role="math" localid="1644859186657" $$\begin{gathered} f\left(x\right)=\frac{24x^2}{2x-4} \\ g\left(x\right)=\frac{12x^2+36x}{x^2-11x+18} \end{gathered}$$

Short answer

The required result is$\frac{x\left(x-9\right)}{\left(x+3\right)}$.

Step-by-step solution

Step 1. Given Information

The given rational expressions are$$\begin{gathered} f\left(x\right)=\frac{24x^2}{2x-4} \\ g\left(x\right)=\frac{12x^2+36x}{x^2-11x+18} \end{gathered}$$

Step 2. Rewrite the Expression as a Product 

• Substitute the given values into $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$.
• $R\left(x\right)=\frac{\frac{24x^2}{2x-4}}{\frac{12x^2+36x}{x^2-11x+18}}$
  
• Write the obtained expression as the product of numerator and the reciprocal of denominator.
• $R\left(x\right)=\frac{24x^2}{2x-4}\times \frac{x^2-11x+18}{12x^2+36x}$
  

Step 3. Factor the Numerator and Denomiantor 

• Express the numerator and denominator of the obtained expression as factors.

$R\left(x\right)=\frac{12\times 2\times x\times x}{2\left(x-2\right)}\times \frac{\left(x-2\right)\left(x-9\right)}{12\times x\times \left(x+3\right)}$

Step 4. Cancel out the Common Factors 

• Simplify the obtained expression by canceling out the common factors.

$\begin{array}{rcl}R\left(x\right)& =& \frac{\cancel{12}\times \cancel{2}\times \cancel{x}\times x}{\cancel{2}\cancel{\left(x-2\right)}}\times \frac{\cancel{\left(x-2\right)}\left(x-9\right)}{\cancel{12}\times \cancel{x}\times \left(x+3\right)}\\ & =& \frac{x\left(x-9\right)}{\left(x+3\right)}\end{array}$

• So, the simplified values of$R\left(x\right)$is$\frac{x\left(x-9\right)}{\left(x+3\right)}$.