poopey
Answered

Which expression is equivalent to [tex]\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}[/tex]?

A. [tex]\frac{c+3}{c^2-4} \div \frac{c+2}{3\left(c^2-9\right)}[/tex]

B. [tex]\frac{c^2-4}{c+3} \div \frac{3\left(c^2-9\right)}{c+2}[/tex]

C. [tex]\frac{c^2-4}{c+3} \cdot \frac{3\left(c^2-9\right)}{c+2}[/tex]

D. [tex]\frac{c+3}{c^2-4} \cdot \frac{c+2}{3\left(c^2-9\right)}[/tex]



Answer :

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].