Let `f(x)=((x-3)(x+2)(x+5))/((x+1)(x-7))`. Find the intervals were f(x) is positive or negative .

Answer

To find the intervals where the function \( f(x) = \frac{(x-3)(x+2)(x+5)}{(x+1)(x-7)} \) is positive or negative, we will follow these steps:

### Step 1: Identify the critical points
The critical points occur where the function is either zero or undefined. This happens at the roots of the numerator and the denominator.

- **Numerator**: Set \( (x-3)(x+2)(x+5) = 0 \)
- \( x - 3 = 0 \) → \( x = 3 \)
- \( x + 2 = 0 \) → \( x = -2 \)
- \( x + 5 = 0 \) → \( x = -5 \)

- **Denominator**: Set \( (x+1)(x-7) = 0 \)
- \( x + 1 = 0 \) → \( x = -1 \)
- \( x - 7 = 0 \) → \( x = 7 \)

So, the critical points are \( x = -5, -2, -1, 3, 7 \).

### Step 2: Create a number line
We will place these critical points on a number line:

```
---|----|----|----|----|----|----|----
-5 -2 -1 3 7
```

### Step 3: Test intervals
We will test the sign of \( f(x) \) in each interval defined by these critical points:

1. **Interval \( (-\infty, -5) \)**:
- Choose \( x = -6 \):
\[ f(-6) = \frac{(-6-3)(-6+2)(-6+5)}{(-6+1)(-6-7)} = \frac{(-9)(-4)(-1)}{(-5)(-13)} = \frac{-36}{65} < 0 \]

2. **Interval \( (-5, -2) \)**:
- Choose \( x = -4 \):
\[ f(-4) = \frac{(-4-3)(-4+2)(-4+5)}{(-4+1)(-4-7)} = \frac{(-7)(-2)(1)}{(-3)(-11)} = \frac{14}{33} > 0 \]

3. **Interval \( (-2, -1) \)**:
- Choose \( x = -1.5 \):
\[ f(-1.5) = \frac{(-1.5-3)(-1.5+2)(-1.5+5)}{(-1.5+1)(-1.5-7)} = \frac{(-4.5)(0.5)(3.5)}{(-0.5)(-8.5)} = \frac{-7.875}{4.25} < 0 \]

4. **Interval \( (-1, 3) \)**:
- Choose \( x = 0 \):
\[ f(0) = \frac{(0-3)(0+2)(0+5)}{(0+1)(0-7)} = \frac{(-3)(2)(5)}{(1)(-7)} = \frac{-30}{-7} > 0 \]

5. **Interval \( (3, 7) \)**:
- Choose \( x = 5 \):
\[ f(5) = \frac{(5-3)(5+2)(5+5)}{(5+1)(5-7)} = \frac{(2)(7)(10)}{(6)(-2)} = \frac{140}{-12} < 0 \]

6. **Interval \( (7, \infty) \)**:
- Choose \( x = 8 \):
\[ f(8) = \frac{(8-3)(8+2)(8+5)}{(8+1)(8-7)} = \frac{(5)(10)(13)}{(9)(1)} = \frac{650}{9} > 0 \]

### Step 4: Summarize the results
From our testing, we find:

- \( f(x) < 0 \) in the intervals: \( (-\infty, -5) \), \( (-2, -1) \), \( (3, 7) \)
- \( f(x) > 0 \) in the intervals: \( (-5, -2) \), \( (-1, 3) \), \( (7, \infty) \)

### Final Answer
The intervals where \( f(x) \) is positive are:
\[ (-5, -2) \cup (-1, 3) \cup (7, \infty) \]
The intervals where \( f(x) \) is negative are:
\[ (-\infty, -5) \cup (-2, -1) \cup (3, 7) \]