Question

Simplify the rational expression.

$\frac{y^2-3y-18}{2y^2+7y+3}$

Short answer

The simplified rational expression is$\frac{y-6}{2y+1}$

Step-by-step solution

Step 1. Apply the concept of simplifying rational expressions.

To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions:

$\frac{AC}{BC}=\frac{A}{B}$

This allows us to cancel common factors from the numerator and denominator.

Step 2. Analyze the expression.

Given$\frac{y^2-3y-18}{2y^2+7y+3}$

This is a rational expression.

To simplify the rational expression, we factor both numerator and denominator and use this$\frac{AC}{BC}=\frac{A}{B}$

Therefore, to simplify a rational expression, we cancel factors that are common to the numerator and denominator.

Step 3. Simplify the expression.

Consider

$\begin{array}{l}\frac{y^2-3y-18}{2y^2+7y+3}=\frac{y^2-6y+3y-18}{2y^2+y+6y+3}\\ \Rightarrow \frac{y^2-3y-18}{2y^2+7y+3}=\frac{y\left(y-6\right)+3\left(y-6\right)}{y\left(2y+1\right)+3\left(2y+1\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\\ \Rightarrow \frac{y^2-3y-18}{2y^2+7y+3}=\frac{\left(y+3\right)\left(y-6\right)}{\left(y+3\right)\left(2y+1\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\end{array}$

The common factor to the numerator and denominator is$y+3$

So we will cancel the common factor from the numerator and denominator.

The simplified expression is$\frac{\left(y-6\right)\left(y+3\right)}{\left(2y+1\right)\left(y+3\right)}=\frac{y-6}{2y+1}$