Question

Complete the expression.
StartFraction 19 Over z squared minus 7 z EndFraction
equalsStartFraction nothing Over z left parenthesis z minus 7 right parenthesis left parenthesis z minus 5 right parenthesis EndFraction
  
​(Simplify your answer. Use factored​ form.)

Answer 1

To complete the expression

\[
\frac{19}{z^2 - 7z} = \frac{ \text{nothing} }{ z (z - 7)(z - 5) }
\]

we first need to factor the denominator on the left side.

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

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

So we rewrite the left side of the equation:

\[
\frac{19}{z(z - 7)} = \frac{\text{nothing}}{z(z - 7)(z - 5)}
\]

To have a common denominator on both sides of the equation, we can multiply the numerator on the left side by \((z - 5)\):

\[
\frac{19(z - 5)}{z(z - 7)(z - 5)} = \frac{\text{nothing}}{z(z - 7)(z - 5)}
\]

Now, we can see that the numerator on the left side is:

\[
19(z - 5)
\]

So the completed expression is:

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

To answer the original question, we can say:

\[
\text{nothing} = 19(z - 5)
\]

Thus, the final expression is:

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

In this expression, you can simplify it:

\[
\frac{19}{z(z - 7)} \quad \text{for } z \neq 5
\]

is the simplified form.