Question

Find the LCD for the expressions $\frac{2}{x^2-x-12},\frac{1}{x^2-16}$rewrite them as equivalent rational expressions with the lowest common denominator.

Short answer

(a) The LCD of the rational expressions is$\left(x+4\right)\left(x+3\right)\left(x-4\right)$

(b) The required rational expressions are $\frac{2\left(x+4\right)}{\left(x-4\right)\left(x+3\right)\left(x+4\right)},\frac{\left(x+3\right)}{\left(x-4\right)\left(x+4\right)\left(x+3\right)}$

Step-by-step solution

Part (a) Step 1. Given information

The given rational expressions are$\frac{2}{x^2-x-12},\frac{1}{x^2-16}$.

Part (a) Step 2. Find LCD

• Factor the denominators of both the expressions.

$\begin{array}{rcl}{x}^2-x-12& =& \left(x-4\right)\left(x+3\right)\\ x^2-16& =& \left(x-4\right)\left(x+4\right)\end{array}$

• So, the LCD will be $(x+3)\left(x-4\right)\left(x+4\right)$

Part (b) Step 1. Given Information

The given expressions are$\frac{2}{x^2-x-12},\frac{1}{x^2-16}$

Part (b) Step 2. Multiply by missing Factors

• Multiply each rational expressions' numerator and denominator by the missing factor.

$$\begin{gathered} \frac{2}{x^2-x-12}=\frac{2\left(x+4\right)}{\left(x-4\right)\left(x+3\right)\left(x+4\right)} \\ \frac{1}{x^2-16}=\frac{\left(x+3\right)}{\left(x-4\right)\left(x+4\right)\left(x+3\right)} \end{gathered}$$