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
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To simplify the product \( \frac{x^{2}-3 x-10}{x^{2}-6 x+5} \cdot \frac{x-2}{x-5} \), we start with factoring both the numerator and denominator of the first fraction. The numerator \( x^2 - 3x - 10 \) factors to \( (x-5)(x+2) \), and the denominator \( x^2 - 6x + 5 \) factors to \( (x-5)(x-1) \). The complete expression then becomes: \[ \frac{(x-5)(x+2)}{(x-5)(x-1)} \cdot \frac{x-2}{x-5} \] Next, we can cancel \( (x-5) \) from the numerator and denominator (noting that \( x \neq 5 \) to avoid division by zero). The simplified version still retains two \( (x-5) \) terms to consider: \[ \frac{(x+2)(x-2)}{(x-1)} \] After multiplying out the numerator, we find: \[ = \frac{x^2 - 4}{x-1} \] Thus, the simplest form of this product has a numerator of \( x^2 - 4 \). As for the excluded values, the original expression has \( x = 5 \) from the denominator and a value from \( x-1 = 0 \), so we also have \( x = 1 \) as an excluded value. Thus, one of the excluded values of \( x \) is 5, and the simplified version has \( \text{x} \text{ values of: } 1 \text{ and } 5 \text{ that cannot be included.} \) So, the answers are: The simplest form of this product has a numerator of \( x^2 - 4 \). The expression has an excluded value of \( x = 5 \) or \( x = 1\).
