Question

In the following exercises, (a) find the LCD for the given rational expressions (b) rewrite them as equivalent rational expressions with the lowest common denominator.

$\frac{8}{y^2+12y+35},\frac{3y}{y^2+y-42}$

Short answer

(a) The LCD is $\left(y+5\right)\left(y-6\right)\left(y+7\right)$.

(b) The expressions with LCD are$\frac{8\left(y-6\right)}{\left(y+5\right)\left(y-6\right)\left(y+7\right)},\frac{3y\left(y+5\right)}{\left(y+5\right)\left(y-6\right)\left(y+7\right)}$.

Step-by-step solution

Part (a) Step 1. Given Information

The given expressions are$\frac{8}{y^2+12y+35},\frac{3y}{y^2+y-42}$.

Part (a) Step 2. Find LCD 

• Factor the denominators of both the given rational expressions.

$\begin{array}{rcl}{y}^2+12y+35& =& \left(y+7\right)\left(y+5\right)\\ y^2+y-42& =& \left(y+7\right)\left(y-6\right)\end{array}$

• So, the LCD is$\left(y+5\right)\left(y-6\right)\left(y+7\right)$.

Part (b) Step 1. Given Information

The given expressions are$\frac{8}{y^2+12y+35},\frac{3y}{y^2+y-42}$.

Part (b) Step 2. Express the expressions with LCD 

• Multiply and divide the expressions by the missing factor to make the denominator equal to LCD.

$$\begin{gathered} \frac{8}{y^2+12y+35}=\frac{8\left(y-6\right)}{\left(y+5\right)\left(y-6\right)\left(y+7\right)} \\ \frac{3y}{y^2+y-42}=\frac{3y\left(y+5\right)}{\left(y+5\right)\left(y-6\right)\left(y+7\right)} \end{gathered}$$

• So, the expressions with equal denominators are$\frac{8\left(y-6\right)}{\left(y+5\right)\left(y-6\right)\left(y+7\right)},\frac{3y\left(y+5\right)}{\left(y+5\right)\left(y-6\right)\left(y+7\right)}$.