Select the correct answer from each drop-down menu. Consider this product. \( \frac{x^{2}-3 x-10}{x^{2}-6 x+5} \cdot \frac{x-2}{x-5} \) The simplest form of this product has a numerator of The expression has an excluded value of \( x= \) Reset

To simplify the given expression and identify the excluded values, follow these steps: ### Given Expression: \[ \frac{x^{2}-3x-10}{x^{2}-6x+5} \cdot \frac{x-2}{x-5} \] ### Step 1: Factor the Numerators and Denominators 1. **Factor \( x^{2}-3x-10 \):** \[ x^{2}-3x-10 = (x - 5)(x + 2) \] 2. **Factor \( x^{2}-6x+5 \):** \[ x^{2}-6x+5 = (x - 1)(x - 5) \] ### Step 2: Rewrite the Expression with Factored Forms \[ \frac{(x - 5)(x + 2)}{(x - 1)(x - 5)} \cdot \frac{x - 2}{x - 5} \] ### Step 3: Cancel Common Factors - **Cancel \( (x - 5) \) from the first fraction:** \[ \frac{(x + 2)}{(x - 1)} \cdot \frac{x - 2}{x - 5} \] - **No further cancellation is possible.** ### Step 4: Simplify the Numerator - Multiply the numerators: \[ (x + 2)(x - 2) = x^{2} - 4 \] ### Simplified Expression: \[ \frac{x^{2} - 4}{(x - 1)(x - 5)} \] ### Excluded Values: Excluded values are the values of \( x \) that make any denominator zero in the original expression. These are: - From \( x^{2}-6x+5 \): \( x = 1 \) and \( x = 5 \) - From \( x - 5 \): \( x = 5 \) **Thus, the excluded values are \( x = 1 \) and \( x = 5 \).** ### **Final Answers:** - **Numerator of the Simplest Form:** \( x^{2} - 4 \) - **Excluded Values:** \( x = 1 \) and \( x = 5 \)