√(х+12)(х-1)(х-9)

To simplify the expression \( \sqrt{(x+12)(x-1)(x-9)} \), we need to analyze the polynomial inside the square root. The expression itself cannot be simplified without specific values for \( x \). However, we can provide insights into its behavior. 1. **Determine the domain**: The expression under the square root must be non-negative for it to be real. Therefore, we set: \[ (x+12)(x-1)(x-9) \geq 0 \] 2. **Identify the critical points**: The critical points where the expression equals zero are: - \( x = -12 \) - \( x = 1 \) - \( x = 9 \) 3. **Test intervals**: The critical points divide the number line into intervals. We can check the sign of the product \( (x+12)(x-1)(x-9) \) in each interval: - Interval \( (-\infty, -12) \) - Interval \( (-12, 1) \) - Interval \( (1, 9) \) - Interval \( (9, \infty) \) 4. **Choose sample points from each interval**: - For \( x < -12 \), let \( x = -13 \): \(((-13)+12)((-13)-1)((-13)-9) = (-1)(-14)(-22) < 0\) (negative) - For \( -12 < x < 1 \), let \( x = 0 \): \((0+12)(0-1)(0-9) = (12)(-1)(-9) > 0\) (positive) - For \( 1 < x < 9 \), let \( x = 5 \): \((5+12)(5-1)(5-9) = (17)(4)(-4) < 0\) (negative) - For \( x > 9 \), let \( x = 10 \): \((10+12)(10-1)(10-9) = (22)(9)(1) > 0\) (positive) 5. **Summarize**: The expression is non-negative in the intervals: - \( x \in [-12, 1] \) (inclusive of endpoints where the product equals zero) - \( x \in [9, \infty) \) 6. **Conclusion**: Thus, the domain of the expression \( \sqrt{(x+12)(x-1)(x-9)} \) is \( x \in [-12, 1] \cup [9, \infty) \). This means when you need to work with this expression, ensure \( x \) falls within these intervals for the square root to be defined, and consider evaluating it for specific values of \( x \) as needed.