Question

For 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 given.

localid="1644858875634" $$\begin{gathered} f\left(x\right)=\frac{27x^2}{3x-21} \\ g\left(x\right)=\frac{3x^2+18x}{x^2+13x+42} \end{gathered}$$

Short answer

The resultant expression is$\frac{3x(x+7)}{(x-7)}$.

Step-by-step solution

Step 1. Given information

The given rational expressions are$$\begin{gathered} f\left(x\right)=\frac{27x^2}{3x-21} \\ g\left(x\right)=\frac{3x^2+18x}{x^2+13x+42} \end{gathered}$$

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="1644817631746" $R\left(x\right)=\frac{{\displaystyle \frac{27x^2}{3x-21}}}{{\displaystyle \frac{3x^2+18x}{x^2+13x+42}}}$

• Write the obtained epression as the product of numerator and the reciprocal of denominator.

$R\left(x\right)=\frac{27x^2}{3x-21}\times \frac{x^2+13x+42}{3x^2+18x}$

Step 3. Factor the Numerator and Denomiantor 

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

$R\left(x\right)=\frac{3\times 3\times 3\times x\times x}{3(x-7)}\times \frac{(x+6)(x+7)}{3x(x+6)}$

Step 4. Cancel out the Common Factors 

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

$\begin{array}{rcl}R\left(x\right)& =& \frac{\cancel{3}\times \cancel{3}\times 3\times \cancel{x}\times x}{\cancel{3}\left(x-7\right)}\times \frac{\cancel{\left(x+6\right)}\left(x+7\right)}{\cancel{3}\cancel{x}\cancel{\left(x+6\right)}}\\ & =& \frac{3x(x+7)}{(x-7)}\end{array}$

• So, the simplified result is$R\left(x\right)=\frac{3x(x+7)}{(x-7)}$.