To find the common denominator of the expression [tex]\(\frac{5}{x^2-4} - \frac{2}{x+2}\)[/tex], let's break down the steps carefully.
1. Factor the expressions in the denominators where possible:
- The denominator [tex]\(x^2 - 4\)[/tex] can be factored. It is a difference of squares and can be written as:
[tex]\[
x^2 - 4 = (x+2)(x-2)
\][/tex]
2. Rewrite each fraction with its factored denominator:
- The term [tex]\(\frac{5}{x^2 - 4}\)[/tex] becomes [tex]\(\frac{5}{(x+2)(x-2)}\)[/tex].
- The term [tex]\(\frac{2}{x+2}\)[/tex] remains the same but we will modify it to have the same denominator as the first term.
3. Find the common denominator:
- The least common denominator (LCD) that can accommodate both denominators [tex]\((x+2)(x-2)\)[/tex] and [tex]\((x+2)\)[/tex] is [tex]\((x+2)(x-2)\)[/tex].
- This is because, unlike in simpler cases of finding the least common multiple of numbers where you'd pick the largest exponent of each prime factor, here you need to have a product that all the denominators can divide into without introducing any new factors beyond those already present in any of the original denominators.
4. Verify the choices given:
- Given the options:
[tex]\((x+2)(x-2)\)[/tex]
[tex]\(x-2\)[/tex]
[tex]\((x+2)^2(x-2)\)[/tex]
[tex]\(x+2\)[/tex]
- The correct common denominator should be a combination that can cover all terms in the denominators of both fractions.
5. Conclusion:
- The common denominator of [tex]\(\frac{5}{x^2-4}-\frac{2}{x+2}\)[/tex] is [tex]\((x+2)(x-2)\)[/tex].
Therefore, the common denominator for the given expression is [tex]\((x+2)(x-2)\)[/tex].
