Question

what is the least common denominator of the expression below?\\(\frac{x^2}{x^2 - 16} + \frac{9 + x}{8x + 2x^2}\\)\\(\bigcirc\\) a. \\((x + 4)\\)\\(\bigcirc\\) b. \\((x + 4)(x - 4)\\)\\(\bigcirc\\) c. \\(2(x + 4)(x - 4)\\)\\(\bigcirc\\) d. \\(2x(x + 4)(x - 4)\\)

Explanation:

Step1: Factor first denominator

Factor \(x^2 - 16\) using difference of squares: \(x^2 - 16=(x + 4)(x - 4)\).

Step2: Factor second denominator

Factor \(8x + 2x^2\): first factor out \(2x\), so \(2x^2+8x = 2x(x + 4)\).

Step3: Find LCD

The least common denominator (LCD) is the product of the highest power of each factor present. The factors are \(2x\), \((x + 4)\), and \((x - 4)\). So LCD is \(2x(x + 4)(x - 4)\).

Answer:

D. \(2x(x + 4)(x - 4)\)