Question

question

find the lcd for \\(\frac{2}{x^2 - x - 12}, \frac{1}{x^2 - 16}\\).

provide your answer below:

Explanation:

Factor the first denominator

Using the Factoring Trinomials knowledge point

$$ x^2 - x - 12 = (x - 4)(x + 3) $$

Factor the second denominator

Using the Difference of Squares knowledge point

$$ x^2 - 16 = (x - 4)(x + 4) $$

Find the Least Common Denominator (LCD)

To find the LCD, we list each unique prime factor with its highest power occurring in either denominator:

• Unique factors: \((x - 4)\), \((x + 3)\), and \((x + 4)\).
• Each factor appears at most once in any single factorization.
• Therefore, the LCD is the product of these unique factors:

$$ \text{LCD} = (x - 4)(x + 3)(x + 4) $$

Answer:

\((x - 4)(x + 3)(x + 4)\)