Question

Write the rational expression as an equivalent expression with the indicated denominator.
StartFraction 18 r Over r squared minus r minus 12 EndFraction equals StartFraction question mark Over left parenthesis r minus 4 right parenthesis left parenthesis r plus 3 right parenthesis left parenthesis r plus 2 right parenthesis EndFraction

Answer 1

To rewrite the expression \(\frac{18r}{r^2 - r - 12}\) with the indicated denominator \((r - 4)(r + 3)(r + 2)\), we first need to factor \(r^2 - r - 12\).

1. **Factor the denominator**:
We are looking for two numbers that multiply to \(-12\) (the constant term) and add to \(-1\) (the coefficient of \(r\)). Those numbers are \(-4\) and \(3\).

So, we can factor:
\[
r^2 - r - 12 = (r - 4)(r + 3)
\]

2. **Rewrite the original rational expression**:
Now we can express the original rational expression:
\[
\frac{18r}{r^2 - r - 12} = \frac{18r}{(r - 4)(r + 3)}
\]

3. **Set the equal sign with the new denominator**:
To adjust it to have the new denominator of \((r - 4)(r + 3)(r + 2)\), we can multiply the numerator and denominator of \(\frac{18r}{(r - 4)(r + 3)}\) by \((r + 2)\):
\[
\frac{18r}{(r - 4)(r + 3)} \cdot \frac{(r + 2)}{(r + 2)} = \frac{18r(r + 2)}{(r - 4)(r + 3)(r + 2)}
\]

4. **Write the final expression**:
This gives us the expression with the specified denominator:
\[
\frac{18r(r + 2)}{(r - 4)(r + 3)(r + 2)}
\]

Thus, the rational expression as an equivalent expression with the indicated denominator is:
\[
\frac{18r(r + 2)}{(r - 4)(r + 3)(r + 2)}
\]
So, the answer is:
\[
18r(r + 2)
\]