Sure, let's simplify the given expression step by step:
Given expression:
[tex]\[
\frac{z^2 - 25}{z} \div \frac{z + 5}{z - 5}
\][/tex]
Step 1: Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we rewrite the expression as:
[tex]\[
\frac{z^2 - 25}{z} \times \frac{z - 5}{z + 5}
\][/tex]
Step 2: Next, we factorize the expression [tex]\( z^2 - 25 \)[/tex]. Notice that [tex]\( z^2 - 25 \)[/tex] is a difference of squares and can be factored as:
[tex]\[
z^2 - 25 = (z - 5)(z + 5)
\][/tex]
Step 3: Substitute the factored form into the given expression:
[tex]\[
\frac{(z - 5)(z + 5)}{z} \times \frac{z - 5}{z + 5}
\][/tex]
Step 4: We can now simplify the expression by canceling common factors in the numerator and the denominator. Cancel out [tex]\( z + 5 \)[/tex] from the numerator and denominator:
[tex]\[
\frac{(z - 5)(z + 5)}{z (z + 5)} \times \frac{z - 5}{z + 5} = \frac{z - 5}{z}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{z - 5}{z}
\][/tex]
So, the final answer is:
[tex]\[
\frac{z - 5}{z}
\][/tex]
