Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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].
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].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.