Question

Perform the indicated operations.
StartFraction 4 Over r squared minus 4 r EndFraction plus StartFraction 4 Over r squared plus 8 r EndFraction minus StartFraction 1 Over r squared plus 4 r minus 32 EndFraction

Answer 1

To perform the operations on the given fractions, let's start by rewriting the expression with a common denominator. The expression is:

\[
\frac{4}{r^2 - 4r} + \frac{4}{r^2 + 8r} - \frac{1}{r^2 + 4r - 32}
\]

First, we will factor the denominators:

1. \( r^2 - 4r = r(r - 4) \)
2. \( r^2 + 8r = r(r + 8) \)
3. To factor \( r^2 + 4r - 32 \), we look for two numbers that multiply to \(-32\) and add to \(4\). These numbers are \(8\) and \(-4\), so:
\[
r^2 + 4r - 32 = (r - 4)(r + 8)
\]

Next, we rewrite our expression with the factored denominators:

\[
\frac{4}{r(r - 4)} + \frac{4}{r(r + 8)} - \frac{1}{(r - 4)(r + 8)}
\]

Now, we need to find the least common denominator (LCD). The LCD will be:

\[
r(r - 4)(r + 8)
\]

Now, we rewrite each fraction with this common denominator:

1. For the first term:
\[
\frac{4}{r(r - 4)} = \frac{4(r + 8)}{r(r - 4)(r + 8)}
\]

2. For the second term:
\[
\frac{4}{r(r + 8)} = \frac{4(r - 4)}{r(r + 8)(r - 4)}
\]

3. For the third term:
\[
\frac{1}{(r - 4)(r + 8)} = \frac{r}{r(r - 4)(r + 8)}
\]

Putting it all together, we have:

\[
\frac{4(r + 8) + 4(r - 4) - r}{r(r - 4)(r + 8)}
\]

Now, simplify the numerator:

\[
4(r + 8) + 4(r - 4) - r = 4r + 32 + 4r - 16 - r = 7r + 16
\]

Thus, we rewrite the expression:

\[
\frac{7r + 16}{r(r - 4)(r + 8)}
\]

This is the simplified form of the original expression. Therefore, the final result is:

\[
\frac{7r + 16}{r(r - 4)(r + 8)}
\]