Find the least common denominator for the following rational expressions. \( \frac{18}{r-4}, \frac{-5 r}{r^{2}-5 r+4}, \frac{r+7}{r^{2}-4 r-5} \)

To find the least common denominator (LCD) of the given rational expressions, we first need to factor the denominators. 1. The first denominator is \( r - 4 \) (already factored). 2. The second denominator is \( r^{2} - 5r + 4 \), which factors to \( (r - 4)(r - 1) \). 3. The third denominator is \( r^{2} - 4r - 5 \), which factors to \( (r - 5)(r + 1) \). Now we list out the different factors from each denominator: - From the first expression: \( r - 4 \). - From the second expression: \( (r - 4)(r - 1) \). - From the third expression: \( (r - 5)(r + 1) \). To create the LCD, we take the highest power of each factor appearing in the factorizations: - The factor \( r - 4 \) appears in the first and second expressions. - The factor \( r - 1 \) appears in the second expression. - The factor \( r - 5 \) appears in the third expression. - The factor \( r + 1 \) appears in the third expression. Thus, the least common denominator is: \[ \text{LCD} = (r - 4)(r - 1)(r - 5)(r + 1) \] Therefore, the least common denominator for the given rational expressions is \( (r - 4)(r - 1)(r - 5)(r + 1) \).