Question

Find the LCD for$\frac{2}{x^2-x-12},\frac{1}{x^2-16}$

Short answer

The required LCD is$\left(x+3\right)\left(x+4\right)\left(x-4\right).$

Step-by-step solution

GIven information

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

Factor the denominator

$\frac{2}{x^2-x-12},\frac{1}{x^2-16}$

Factorize the expression

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

$\left(x^2-16\right)=\left(x+4\right)\left(x-4\right)$

Multiply all the distinct factors of the expressions $=\left(x-4\right)\left(x+3\right)\left(x+4\right)$

Therefore the LCD of rational expressions isrole="math" localid="1648189296444" $\left(x+3\right)\left(x+4\right)\left(x-4\right)$.