WARM-UP Find the least common denominator for the following rational expressions. $$ \frac{18}{r-4} \cdot \frac{-5 r}{r^{2}-5 r+4} \cdot \frac{r+7}{r^{2}-4 r-5} $$ $$ \begin{array}{l}(r-1)(r-4)(r-5) \ \mathbf{x}(r-1)(r+4)(r-5) \ (r-1)(r+1)(r-4)(r-5) \ (r-1)(r+1)(r+4)(r-5)\end{array} $$

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UpStudy AI · Accepted Answer

We begin by factoring each denominator.

1. The first denominator is  
   $$
   r-4.
   $$

2. The second denominator is  
   $$
   r^2-5r+4.
   $$  
   To factor this, we look for two numbers that multiply to $$4$$ and add to $$-5$$. The numbers $$-1$$ and $$-4$$ work since  
   $$
   -1 \times -4 = 4 \quad \text{and} \quad -1 + (-4) = -5.
   $$  
   Thus,  
   $$
   r^2-5r+4 = (r-1)(r-4).
   $$

3. The third denominator is  
   $$
   r^2-4r-5.
   $$  
   Here, we need two numbers that multiply to $$-5$$ and add to $$-4$$. The numbers $$-5$$ and $$1$$ satisfy these conditions since  
   $$
   -5 \times 1 = -5 \quad \text{and} \quad -5+1 = -4.
   $$  
   Therefore,  
   $$
   r^2-4r-5 = (r-5)(r+1).
   $$

Now, the list of distinct factors from all denominators is:
- $$r-4$$ (from the first and second expressions)
- $$r-1$$ (from the second expression)
- $$r-5$$ (from the third expression)
- $$r+1$$ (from the third expression)

The least common denominator (LCD) is the product of all these factors:

$$
(r-1)(r+1)(r-4)(r-5).
$$

Among the given choices, this corresponds to the third option:

$$
(r-1)(r+1)(r-4)(r-5).
$$
The least common denominator is $$ (r-1)(r+1)(r-4)(r-5) $$.

WARM-UP Find the least common denominator for the following rational expressions. \( \frac{18}{r-4} \cdot \frac{-5 r}{r^{2}-5 r+4} \cdot \frac{r+7}{r^{2}-4 r-5} \) \( \begin{array}{l}(r-1)(r-4)(r-5) \\ \mathbf{x}(r-1)(r+4)(r-5) \\ (r-1)(r+1)(r-4)(r-5) \\ (r-1)(r+1)(r+4)(r-5)\end{array} \)

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