Question
Find the least common denominator for the following pair of rational expressions.
StartFraction 4 Over k squared plus 7 k EndFraction
and StartFraction 3 Over k squared plus 4 k minus 21 EndFraction
StartFraction 4 Over k squared plus 7 k EndFraction
and StartFraction 3 Over k squared plus 4 k minus 21 EndFraction
Answers
GPT-4o mini
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.
1. **Factoring the first denominator, \(k^2 + 7k\):**
We can factor out a common term \(k\):
\[
k^2 + 7k = k(k + 7).
\]
2. **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: \((k + 7)(k - 3)\)
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
\[
k(k + 7)(k - 3).
\]
\[
\frac{4}{k^2 + 7k} \quad \text{and} \quad \frac{3}{k^2 + 4k - 21},
\]
we need to factor each denominator.
1. **Factoring the first denominator, \(k^2 + 7k\):**
We can factor out a common term \(k\):
\[
k^2 + 7k = k(k + 7).
\]
2. **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: \((k + 7)(k - 3)\)
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
\[
k(k + 7)(k - 3).
\]