√(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.
- Determine the domain: The expression under the square root must be non-negative for it to be real. Therefore, we set:
\[ (x+12)(x-1)(x-9) \geq 0 \]
Identify the critical points: The critical points where the expression equals zero are:
- \( x = -12 \)
- \( x = 1 \)
- \( x = 9 \)
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 \( (9, \infty) \)
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)
Summarize: The expression is non-negative in the intervals:
- \( x \in [-12, 1] \) (inclusive of endpoints where the product equals zero)
- \( x \in [9, \infty) \)
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.
