To find the least common denominator (LCD) for the given rational expressions, we need to identify the denominators of each expression and then determine the least common multiple of those denominators.
The denominators provided are:
- \((x + 2)(x - 4)\)
- \(x(x - 4)\)
- \(x(x + 2)(x - 4)\)
Now, let's break each denominator down into its factors:
- The first denominator \((x + 2)(x - 4)\) consists of the factors \(x + 2\) and \(x - 4\).
- The second denominator \(x(x - 4)\) consists of the factors \(x\) and \(x - 4\).
- The third denominator \(x(x + 2)(x - 4)\) consists of the factors \(x\), \(x + 2\), and \(x - 4\).
Now, to find the LCD, we take each distinct factor at its highest power across all denominators. The distinct factors are:
- \(x\)
- \(x + 2\)
- \(x - 4\)
Therefore, the least common denominator (LCD) will include each of these factors:
The LCD is: \[ x(x + 2)(x - 4) \]
This is the least common denominator for the given rational expressions.