Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which expression is equivalent to \(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\), we need to perform the division operation by transforming it into a multiplication by the reciprocal.
1. Rewrite the division as a multiplication:
[tex]\[ \frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)} \quad \Rightarrow \quad \frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2} \][/tex]
2. Factorize the polynomials where possible:
- \(c^2-4\) is a difference of squares: \(c^2 - 4 = (c - 2)(c + 2)\)
- \(c^2 - 9\) is also a difference of squares: \(c^2 - 9 = (c - 3)(c + 3)\)
Thus, the expression becomes:
[tex]\[ \frac{(c-2)(c+2)}{c+3} \cdot \frac{3(c-3)(c+3)}{c+2} \][/tex]
3. Simplify the expression by canceling common factors:
[tex]\[ \frac{(c-2)(c+2)}{c+3} \cdot \frac{3(c-3)(c+3)}{c+2} = \frac{(c-2)\cancel{(c+2)}}{\cancel{c+3}} \cdot \frac{3(c-3)\cancel{(c+3)}}{\cancel{c+2}} \][/tex]
After canceling the common factors \((c+2)\) and \((c+3)\), we are left with:
[tex]\[ (c-2) \cdot 3(c-3) = 3(c-2)(c-3) \][/tex]
4. Conclude the simplified expression:
[tex]\[ 3(c-2)(c-3) \][/tex]
Now, we need to see which option matches our simplified form. We previously transformed the division into a multiplication:
Comparing with the given options:
1. \(\frac{c+3}{c^2-4} \div \frac{c+2}{3(c^2-9)}\) is incorrect.
2. \(\frac{c^2-4}{c+3} \div \frac{3(c^2-9)}{c+2}\) is incorrect.
3. \(\frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}\) is correct (this matches our step).
4. \(\frac{c+3}{c^2-4} \cdot \frac{c+2}{3(c^2-9)}\) is incorrect.
Thus, the correct equivalent expression is:
[tex]\(\frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}\)[/tex].
1. Rewrite the division as a multiplication:
[tex]\[ \frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)} \quad \Rightarrow \quad \frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2} \][/tex]
2. Factorize the polynomials where possible:
- \(c^2-4\) is a difference of squares: \(c^2 - 4 = (c - 2)(c + 2)\)
- \(c^2 - 9\) is also a difference of squares: \(c^2 - 9 = (c - 3)(c + 3)\)
Thus, the expression becomes:
[tex]\[ \frac{(c-2)(c+2)}{c+3} \cdot \frac{3(c-3)(c+3)}{c+2} \][/tex]
3. Simplify the expression by canceling common factors:
[tex]\[ \frac{(c-2)(c+2)}{c+3} \cdot \frac{3(c-3)(c+3)}{c+2} = \frac{(c-2)\cancel{(c+2)}}{\cancel{c+3}} \cdot \frac{3(c-3)\cancel{(c+3)}}{\cancel{c+2}} \][/tex]
After canceling the common factors \((c+2)\) and \((c+3)\), we are left with:
[tex]\[ (c-2) \cdot 3(c-3) = 3(c-2)(c-3) \][/tex]
4. Conclude the simplified expression:
[tex]\[ 3(c-2)(c-3) \][/tex]
Now, we need to see which option matches our simplified form. We previously transformed the division into a multiplication:
Comparing with the given options:
1. \(\frac{c+3}{c^2-4} \div \frac{c+2}{3(c^2-9)}\) is incorrect.
2. \(\frac{c^2-4}{c+3} \div \frac{3(c^2-9)}{c+2}\) is incorrect.
3. \(\frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}\) is correct (this matches our step).
4. \(\frac{c+3}{c^2-4} \cdot \frac{c+2}{3(c^2-9)}\) is incorrect.
Thus, the correct equivalent expression is:
[tex]\(\frac{c^2-4}{c+3} \cdot \frac{3(c^2-9)}{c+2}\)[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.