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.

$\frac{3}{5m^2-3m-2},\frac{6m}{5m^2+17m+6}$

Short answer

(a) The LCD is $(5m+2)(m-1)(m+3)$.

(b) The expressions with LCD are$\frac{3(m+3)}{(5m+2)(m-1)(m+3)},\frac{6m(m-1)}{(5m+2)(m+3)(m-1)}$.

Step-by-step solution

Part (a) Step 1. Given Information

The given expressions are$\frac{3}{5m^2-3m-2},\frac{6m}{5m^2+17m+6}$.

Part (a) Step 2. Find LCD  

• Factor the denominators of both the given rational expressions.

$$\begin{gathered} 5m^2-3m-2=(5m+2)(m-1) \\ 5m^2+17m+6=(5m+2)(m+3) \end{gathered}$$

• So, the LCD is $(5m+2)(m-1)(m+3)$.

Part (b) Step 1. Given Information

The given expressions are$\frac{3}{5m^2-3m-2},\frac{6m}{5m^2+17m+6}$.

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.

$$\begin{gathered} \frac{3}{5m^2-3m-2}=\frac{3(m+3)}{(5m+2)(m-1)(m+3)} \\ \frac{6m}{5m^2+17m+6}=\frac{6m(m-1)}{(5m+2)(m+3)(m-1)} \end{gathered}$$

• So, the expressions with equal denominators are $\frac{3(m+3)}{(5m+2)(m-1)(m+3)},\frac{6m(m-1)}{(5m+2)(m+3)(m-1)}$.