Question

In the following exercises, find $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$where $f\left(x\right)$and $g\left(x\right)$are known.

$$\begin{gathered} f\left(x\right)=\frac{16x^2}{4x+36} \\ g\left(x\right)=\frac{4x^2-24x}{x^2+4x-45} \end{gathered}$$

Short answer

The resultant expression is$\frac{x\left(x-5\right)}{\left(x-6\right)}$

Step-by-step solution

Step 1. Given Information

The given rational expresions are$$\begin{gathered} f\left(x\right)=\frac{16x^2}{4x+36} \\ g\left(x\right)=\frac{4x^2-24x}{x^2+4x-45} \end{gathered}$$

Step 2. Rewrite the Expression as a Product 

• Substitute the given values into $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$.
• role="math" localid="1644819781853" $R\left(x\right)=\frac{{\displaystyle \frac{16x^2}{4x+36}}}{{\displaystyle \frac{4x^2-24x}{x^2+4x-45}}}$
• Write the obtained expression as the product of numerator and the reciprocal of denominator.
• $R\left(x\right)=\frac{16x^2}{4x+36}\times \frac{x^2+4x-45}{4x^2-24x}$
  

Step 3. Factor the Numerator and Denomiantor  

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

$R\left(x\right)=\frac{4\times 4\times x\times x}{4\left(x+9\right)}\times \frac{\left(x+9\right)\left(x-5\right)}{4\times x\times \left(x-6\right)}$

Step 4. Cancel out the Common Factors  

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

$\begin{array}{rcl}R\left(x\right)& =& \frac{\cancel{4}\times \cancel{4}\times \cancel{x}\times x}{\cancel{4}\cancel{\left(x+9\right)}}\times \frac{\cancel{\left(x+9\right)}\left(x-5\right)}{\cancel{4}\times \cancel{x}\times \left(x-6\right)}\\ & =& \frac{x\left(x-5\right)}{\left(x-6\right)}\end{array}$

• So, the simplified result is$\frac{x\left(x-5\right)}{\left(x-6\right)}$.