Question

Write the rational expression with the indicated denominator. 19/(z^2-7z)=?/z(z-7)(z-5)

Answer 1

To write the rational expression \( \frac{19}{z^2 - 7z} \) with the denominator \( z(z - 7)(z - 5) \), we first need to factor the original denominator.

The expression \( z^2 - 7z \) can be factored as follows:

\[
z^2 - 7z = z(z - 7)
\]

Now, we need to convert \( \frac{19}{z(z - 7)} \) into a form where the denominator is \( z(z - 7)(z - 5) \). We can do this by multiplying both the numerator and the denominator by \( (z - 5) \):

\[
\frac{19}{z(z - 7)} \cdot \frac{(z - 5)}{(z - 5)} = \frac{19(z - 5)}{z(z - 7)(z - 5)}
\]

Now simplifying the numerator:

\[
19(z - 5) = 19z - 95
\]

Putting it all together, we have:

\[
\frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{19z - 95}{z(z - 7)(z - 5)}
\]

Thus, the rational expression with the indicated denominator is:

\[
\frac{19z - 95}{z(z - 7)(z - 5)}
\]