To find the Least Common Denominator (LCD) of the given rational expressions [tex]\(\frac{2}{q^2 - 3q - 54}\)[/tex] and [tex]\(\frac{8q}{q^2 - 11q + 18}\)[/tex], we follow these steps:
1. Factor the denominators of each expression.
2. Identify the highest power of each distinct factor that appears in the denominators.
3. Multiply these factors together to find the LCD.
### Step 1: Factor the denominators
First, we factor the quadratic expressions in the denominators.
For [tex]\(\frac{2}{q^2 - 3q - 54}\)[/tex]:
We need to factor the quadratic expression [tex]\(q^2 - 3q - 54\)[/tex].
To factor it, we look for two numbers that multiply to [tex]\(-54\)[/tex] and add up to [tex]\(-3\)[/tex].
These numbers are [tex]\(-9\)[/tex] and [tex]\(6\)[/tex].
[tex]\[
q^2 - 3q - 54 = (q - 9)(q + 6)
\][/tex]
So, the denominator of the first expression factors to [tex]\((q - 9)(q + 6)\)[/tex].
For [tex]\(\frac{8q}{q^2 - 11q + 18}\)[/tex]:
We need to factor the quadratic expression [tex]\(q^2 - 11q + 18\)[/tex].
To factor it, we look for two numbers that multiply to [tex]\(18\)[/tex] and add up to [tex]\(-11\)[/tex].
These numbers are [tex]\(-9\)[/tex] and [tex]\(-2\)[/tex].
[tex]\[
q^2 - 11q + 18 = (q - 9)(q - 2)
\][/tex]
So, the denominator of the second expression factors to [tex]\((q - 9)(q - 2)\)[/tex].
### Step 2: Identify the highest power of each distinct factor
We now have the factored forms of the denominators:
- [tex]\((q - 9)(q + 6)\)[/tex]
- [tex]\((q - 9)(q - 2)\)[/tex]
The distinct factors are [tex]\(q - 9\)[/tex], [tex]\(q + 6\)[/tex], and [tex]\(q - 2\)[/tex].
The highest power of each factor present in the denominators is:
- [tex]\(q - 9\)[/tex] appears once in both denominators.
- [tex]\(q + 6\)[/tex] appears once in the first denominator.
- [tex]\(q - 2\)[/tex] appears once in the second denominator.
### Step 3: Multiply these factors to find the LCD
The Least Common Denominator is the product of these highest powers:
[tex]\[
\text{LCD} = (q - 9)(q + 6)(q - 2)
\][/tex]
### Final Answer:
The Least Common Denominator (LCD) of the given expressions [tex]\(\frac{2}{q^2 - 3q - 54}\)[/tex] and [tex]\(\frac{8q}{q^2 - 11q + 18}\)[/tex] in factored form is:
[tex]\[
\boxed{(q - 9)(q + 6)(q - 2)}
\][/tex]
