Question

Find the least common denominator for the following trio of rational expressions.
StartFraction 5 Over p squared plus 10 p plus 24 EndFraction

StartFraction 3 Over p squared plus 2 p minus 8 EndFraction
StartFraction 9 Over p squared plus 4 p minus 12 EndFraction

Answer 1

To find the least common denominator (LCD) for the rational expressions, we first need to factor each of the denominators.

1. **First denominator**: \(p^2 + 10p + 24\)

We can factor this quadratic expression:
\[
p^2 + 10p + 24 = (p + 6)(p + 4)
\]

2. **Second denominator**: \(p^2 + 2p - 8\)

We can factor this quadratic expression:
\[
p^2 + 2p - 8 = (p + 4)(p - 2)
\]

3. **Third denominator**: \(p^2 + 4p - 12\)

This quadratic can be factored as well:
\[
p^2 + 4p - 12 = (p + 6)(p - 2)
\]

Now we have the factorizations of the denominators:
- \(p^2 + 10p + 24 = (p + 6)(p + 4)\)
- \(p^2 + 2p - 8 = (p + 4)(p - 2)\)
- \(p^2 + 4p - 12 = (p + 6)(p - 2)\)

Next, we will identify all the unique factors from these expressions:
- From the first denominator: \(p + 6\), \(p + 4\)
- From the second denominator: \(p + 4\), \(p - 2\)
- From the third denominator: \(p + 6\), \(p - 2\)

The unique factors are: \(p + 6\), \(p + 4\), and \(p - 2\).

To make the least common denominator (LCD), we multiply all the unique factors together:
\[
\text{LCD} = (p + 6)(p + 4)(p - 2)
\]

Thus, the least common denominator for the rational expressions is:
\[
\boxed{(p + 6)(p + 4)(p - 2)}
\]