Question

find the least common denominator of \\(\frac{8}{x^2 - 9x - 22}\\) and \\(\frac{-2}{x^2 + 2x}\\).

Explanation:

Step1: Factor the denominators

First, factor \(x^2 - 9x - 22\). We need two numbers that multiply to \(-22\) and add to \(-9\). The numbers are \(-11\) and \(2\), so \(x^2 - 9x - 22=(x - 11)(x + 2)\).
Next, factor \(x^2 + 2x\). We can factor out an \(x\), so \(x^2 + 2x = x(x + 2)\).

Step2: Identify the least common denominator (LCD)

The LCD of two fractions is the least common multiple (LCM) of their denominators. The denominators are \((x - 11)(x + 2)\) and \(x(x + 2)\).
To find the LCM, we take the highest power of each unique factor. The unique factors are \(x\), \(x - 11\), and \(x + 2\). So the LCM (LCD) is \(x(x - 11)(x + 2)\).

Answer:

The least common denominator is \(x(x - 11)(x + 2)\) (or expanded as \(x^3 - 9x^2 - 22x\)).