Column A: \(3x^2\)
Column B: \(3x^3\)
Column A: \(3x^2\)
Column B: \(3x^3\)
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
This question tests the properties of negative numbers raised to even and odd powers.
Step 2: Detailed Explanation:
We are given that \(x\) is a negative number (\(x < 0\)).
Let's analyze Column A: \(3x^2\).
When a negative number (\(x\)) is squared, the result is positive. For example, if \(x = -2\), then: \[ x^2 = (-2)^2 = 4 \] So, \(x^2\) is always positive. A positive number multiplied by 3 is also positive. Thus, \(3x^2\) is positive.
Let's analyze Column B: \(3x^3\).
When a negative number (\(x\)) is cubed (raised to an odd power), the result is negative. For example, if \(x = -2\), then: \[ x^3 = (-2)^3 = -8 \] So, \(x^3\) is always negative. A negative number multiplied by 3 is also negative. Thus, \(3x^3\) is negative.
Step 3: Comparing the Quantities:
Column A (\(3x^2\)) is a positive number.
Column B (\(3x^3\)) is a negative number.
Any positive number is greater than any negative number.
Therefore, the quantity in Column A is greater.
Final Answer:
\[ \boxed{\text{The quantity in Column A is greater.}} \]
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