To find the least common denominator (LCD) for the rational expressions
\[ \frac{4}{k^2 + 7k} \quad \text{and} \quad \frac{3}{k^2 + 4k - 21}, \]
we need to factor each denominator.
-
Factoring the first denominator, \(k^2 + 7k\):
We can factor out a common term \(k\): \[ k^2 + 7k = k(k + 7). \]
-
Factoring the second denominator, \(k^2 + 4k - 21\):
We need to find two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of \(k\)). The numbers \(7\) and \(-3\) fit this requirement: \[ k^2 + 4k - 21 = (k + 7)(k - 3). \]
Now, we have the following factorizations:
- The first denominator: \(k(k + 7)\)
- The second denominator: \((k + 7)(k - 3)\)
- Finding the least common denominator:
The LCD is obtained by taking each distinct factor from both denominators at its highest power:
- From \(k(k + 7)\), we take \(k\) and \(k + 7\).
- From \((k + 7)(k - 3)\), we take \(k - 3\).
Combining these, the least common denominator (LCD) is: \[ \text{LCD} = k(k + 7)(k - 3). \]
Thus, the least common denominator for the given expressions is
\[ k(k + 7)(k - 3). \]