To find the least common denominator (LCD) for the given rational expressions, we need to consider the denominators of each expression. Specifically, we have:
1. [tex]\(\frac{2}{y^2 - 3y - 10}\)[/tex]
2. [tex]\(\frac{6}{y^2 + 8y + 12}\)[/tex]
First, we should factor each quadratic expression in the denominators.
Step 1: Factor [tex]\(y^2 - 3y - 10\)[/tex]
We look for two numbers that multiply to [tex]\(-10\)[/tex] and add to [tex]\(-3\)[/tex]. These numbers are [tex]\(-5\)[/tex] and [tex]\(2\)[/tex].
Thus,
[tex]\[ y^2 - 3y - 10 = (y - 5)(y + 2) \][/tex]
Step 2: Factor [tex]\(y^2 + 8y + 12\)[/tex]
We look for two numbers that multiply to [tex]\(12\)[/tex] and add to [tex]\(8\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(2\)[/tex].
Thus,
[tex]\[ y^2 + 8y + 12 = (y + 6)(y + 2) \][/tex]
Step 3: Determine the Least Common Denominator
The least common denominator must include all distinct factors from both denominators the maximum number of times they appear in any single factored form. Here, the distinct factors are [tex]\( (y - 5) \)[/tex], [tex]\( (y + 2) \)[/tex], and [tex]\( (y + 6) \)[/tex].
The LCD must include:
- [tex]\( (y - 5) \)[/tex] (appears once in [tex]\( (y - 5)(y + 2) \)[/tex])
- [tex]\( (y + 2) \)[/tex] (appears once in both [tex]\( (y - 5)(y + 2) \)[/tex] and [tex]\( (y + 6)(y + 2) \)[/tex])
- [tex]\( (y + 6) \)[/tex] (appears once in [tex]\( (y + 6)(y + 2) \)[/tex])
Therefore, the least common denominator (LCD) is:
[tex]\[ (y - 5)(y + 2)(y + 6) \][/tex]
Summary:
The least common denominator for the rational expressions [tex]\(\frac{2}{y^2 - 3y - 10}\)[/tex] and [tex]\(\frac{6}{y^2 + 8y + 12}\)[/tex] is:
[tex]\[ (y - 5)(y + 2)(y + 6) \][/tex]
