To solve the division of the two given algebraic fractions:
[tex]\[
\frac{z^2 - 25}{z} \div \frac{z + 5}{z - 5}
\][/tex]
We start by remembering that dividing by a fraction is equivalent to multiplying by the reciprocal of that fraction. So, we can rewrite the expression as:
[tex]\[
\frac{z^2 - 25}{z} \times \frac{z - 5}{z + 5}
\][/tex]
Next, we simplify the numerator [tex]\(z^2 - 25\)[/tex]. This is a difference of squares and can be factored as:
[tex]\[
z^2 - 25 = (z + 5)(z - 5)
\][/tex]
Now we substitute this factorized form back into our expression:
[tex]\[
\frac{(z + 5)(z - 5)}{z} \times \frac{z - 5}{z + 5}
\][/tex]
Now, let's multiply these fractions together. When we multiply fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{(z + 5)(z - 5)}{z} \times \frac{z - 5}{z + 5} = \frac{(z + 5)(z - 5)(z - 5)}{z(z + 5)}
\][/tex]
In this fraction, we have factors of [tex]\((z + 5)\)[/tex] in the numerator and the denominator, so we can cancel them out:
[tex]\[
\frac{(z - 5)(z - 5)}{z} = \frac{(z - 5)^2}{z}
\][/tex]
Expanding [tex]\((z - 5)^2\)[/tex], we get:
[tex]\[
(z - 5)^2 = z^2 - 10z + 25
\][/tex]
Thus, our expression simplifies to:
[tex]\[
\frac{z^2 - 10z + 25}{z}
\][/tex]
This can be divided term-by-term:
[tex]\[
\frac{z^2}{z} - \frac{10z}{z} + \frac{25}{z}
\][/tex]
Simplifying each term:
[tex]\[
z - 10 + \frac{25}{z}
\][/tex]
So, the simplified result of the given division is:
[tex]\[
z - 10 + \frac{25}{z}
\][/tex]
