To find the Least Common Denominator (LCD) of the given expressions:
[tex]\[ \frac{4}{n^2 + 5n - 36} \][/tex]
[tex]\[ \frac{7n}{n^2 - 7n + 12} \][/tex]
we follow these steps:
1. Factorize each denominator separately.
First denominator: [tex]\( n^2 + 5n - 36 \)[/tex]
To factorize this quadratic expression, we need to find two numbers that multiply to [tex]\(-36\)[/tex] (the constant term) and add to [tex]\(5\)[/tex] (the coefficient of [tex]\(n\)[/tex]).
These numbers are [tex]\(9\)[/tex] and [tex]\(-4\)[/tex]:
[tex]\[
n^2 + 5n - 36 = (n + 9)(n - 4)
\][/tex]
So, the first denominator factors as:
[tex]\[
n^2 + 5n - 36 = (n - 4)(n + 9)
\][/tex]
Second denominator: [tex]\( n^2 - 7n + 12 \)[/tex]
To factorize this quadratic expression, we need to find two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add to [tex]\(-7\)[/tex] (the coefficient of [tex]\(n\)[/tex]).
These numbers are [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex]:
[tex]\[
n^2 - 7n + 12 = (n - 3)(n - 4)
\][/tex]
So, the second denominator factors as:
[tex]\[
n^2 - 7n + 12 = (n - 4)(n - 3)
\][/tex]
2. Identify all unique factors from both denominators.
For the denominators:
[tex]\[
n^2 + 5n - 36 = (n - 4)(n + 9)
\][/tex]
and
[tex]\[
n^2 - 7n + 12 = (n - 4)(n - 3)
\][/tex]
The unique factors are:
- [tex]\( (n - 4) \)[/tex]
- [tex]\( (n + 9) \)[/tex]
- [tex]\( (n - 3) \)[/tex]
3. Construct the Least Common Denominator (LCD).
The LCD is the product of the highest powers of all unique factors identified. Since we have:
[tex]\[
n^2 + 5n - 36 = (n - 4)(n + 9)
\][/tex]
and
[tex]\[
n^2 - 7n + 12 = (n - 4)(n - 3)
\][/tex]
The LCD must include each unique factor, appearing as many times as it does in any one denominator.
Therefore, we include each factor [tex]\( (n - 4) \)[/tex], [tex]\( (n + 9) \)[/tex], and [tex]\( (n - 3) \)[/tex]:
[tex]\[
\text{LCD} = (n - 4)(n - 3)(n + 9)
\][/tex]
In conclusion, the Least Common Denominator (LCD) of the given expressions is:
[tex]\[
\boxed{(n - 4)(n - 3)(n + 9)}
\][/tex]
