To perform the operation \(\frac{z^2 - 2z - 35}{z^2 - 10z + 25} \div \frac{z - 7}{z - 5}\), we first rewrite the division as multiplication by the reciprocal:
\[ \frac{z^2 - 2z - 35}{z^2 - 10z + 25} \cdot \frac{z - 5}{z - 7} \]
Next, we can factor the quadratics in the expression.
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Factor \(z^2 - 2z - 35\): We need two numbers that add up to \(-2\) and multiply to \(-35\). These numbers are \(5\) and \(-7\). Thus, \[ z^2 - 2z - 35 = (z - 7)(z + 5) \]
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Factor \(z^2 - 10z + 25\): This is a perfect square trinomial that factors to: \[ z^2 - 10z + 25 = (z - 5)^2 \]
Now we substitute the factored forms into the expression:
\[ \frac{(z - 7)(z + 5)}{(z - 5)^2} \cdot \frac{z - 5}{z - 7} \]
Next, we can simplify the expression:
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Cancel \(z - 7\) from the numerator and denominator: \[ \frac{(z + 5)}{(z - 5)} \cdot 1 = \frac{z + 5}{z - 5} \]
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Cancel \(z - 5\): Substitute \(z - 5\) from the first part of the expression. The whole fraction now simplifies to: \[ \frac{z + 5}{z - 5} \]
So, the final result is:
\[ \frac{z + 5}{z - 5} \]
This represents the simplified form of the original operation. Note that \(z \neq 5\) and \(z \neq 7\) to avoid division by zero in the original expressions.