If x is positive and x > y, which of the following must be positive? x^2y - xy^2 x^3 -...

1. Translate the problem requirements: We know \(\mathrm{x}\) is positive (\(\mathrm{x} > 0\)) and \(\mathrm{x} > \mathrm{y}\). We need to determine which of the three expressions must always be positive under these conditions.
2. Factor each expression to reveal underlying relationships: Break down each expression into factors to better understand when they're positive or negative.
3. Analyze the sign of each factored expression: Use the constraints \(\mathrm{x} > 0\) and \(\mathrm{x} > \mathrm{y}\) to determine whether each factor is positive, negative, or could be either.
4. Test with concrete examples: Verify conclusions by testing specific values that satisfy the constraints but explore different scenarios (like when y is positive vs negative).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given and what we need to find.

We know two things about our variables:

• \(\mathrm{x}\) is positive (this means \(\mathrm{x} > 0\))
• \(\mathrm{x}\) is greater than \(\mathrm{y}\) (this means \(\mathrm{x} > \mathrm{y}\))

Notice that we don't know whether \(\mathrm{y}\) is positive or negative - we only know that whatever \(\mathrm{y}\) is, \(\mathrm{x}\) is bigger than it.

We need to figure out which of the three expressions will ALWAYS be positive, no matter what specific values \(\mathrm{x}\) and \(\mathrm{y}\) take (as long as they follow our rules).

Process Skill: TRANSLATE - Converting the constraint language into clear mathematical understanding

2. Factor each expression to reveal underlying relationships

Instead of trying to work with these expressions as they are, let's break them down into simpler pieces. This is like taking apart a machine to see how it works.

Expression I: \(\mathrm{x}^2\mathrm{y} - \mathrm{x}\mathrm{y}^2\)
We can factor out \(\mathrm{x}\mathrm{y}\) from both terms:
\(\mathrm{x}^2\mathrm{y} - \mathrm{x}\mathrm{y}^2 = \mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y})\)

Expression II: \(\mathrm{x}^3 - \mathrm{x}^2\mathrm{y}\)
We can factor out \(\mathrm{x}^2\) from both terms:
\(\mathrm{x}^3 - \mathrm{x}^2\mathrm{y} = \mathrm{x}^2(\mathrm{x} - \mathrm{y})\)

Expression III: \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2\)
We can factor out \(\mathrm{x}\) from both terms:
\(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2 = \mathrm{x}(\mathrm{x}^2 - \mathrm{y}^2)\)
We can factor further since \(\mathrm{x}^2 - \mathrm{y}^2\) is a difference of squares:
\(\mathrm{x}(\mathrm{x}^2 - \mathrm{y}^2) = \mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y})\)

Now our expressions look much cleaner and we can see the patterns!

3. Analyze the sign of each factored expression

Now let's figure out whether each factored expression is positive or negative by looking at each factor.

Remember: \(\mathrm{x} > 0\) and \(\mathrm{x} > \mathrm{y}\)

For Expression I: \(\mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y})\)

• \(\mathrm{x}\) is positive (given)
• \((\mathrm{x} - \mathrm{y})\) is positive (since \(\mathrm{x} > \mathrm{y}\))
• But what about \(\mathrm{y}\)? We don't know if \(\mathrm{y}\) is positive or negative!

If \(\mathrm{y}\) is positive: \(\mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y}) = (+)(+)(+) = \text{positive}\)
If \(\mathrm{y}\) is negative: \(\mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y}) = (+)(-)(+) = \text{negative}\)

So Expression I is not always positive.

For Expression II: \(\mathrm{x}^2(\mathrm{x} - \mathrm{y})\)

• \(\mathrm{x}^2\) is always positive (since \(\mathrm{x} > 0\), and any positive number squared is positive)
• \((\mathrm{x} - \mathrm{y})\) is positive (since \(\mathrm{x} > \mathrm{y}\))

So: \(\mathrm{x}^2(\mathrm{x} - \mathrm{y}) = (+)(+) = \text{positive ALWAYS}\)

For Expression III: \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y})\)

• \(\mathrm{x}\) is positive (given)
• \((\mathrm{x} - \mathrm{y})\) is positive (since \(\mathrm{x} > \mathrm{y}\))
• But what about \((\mathrm{x} + \mathrm{y})\)? This depends on whether \(\mathrm{y}\) is positive or negative!

If \(\mathrm{y}\) is positive: \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y}) = (+)(+)(+) = \text{positive}\)
If \(\mathrm{y}\) is negative and \(|\mathrm{y}| < \mathrm{x}\): \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y}) = (+)(+)(+) = \text{positive}\)
If \(\mathrm{y}\) is negative and \(|\mathrm{y}| > \mathrm{x}\): \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y}) = (+)(+)(-) = \text{negative}\)

So Expression III is not always positive.

Process Skill: CONSIDER ALL CASES - Testing different scenarios for y to ensure we don't miss any possibilities

4. Test with concrete examples

Let's verify our reasoning with specific numbers.

Case 1: \(\mathrm{x} = 3, \mathrm{y} = 1\) (both positive, \(\mathrm{x} > \mathrm{y}\))

• Expression I: \(\mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y}) = (3)(1)(3 - 1) = 3(2) = 6\) ✓ positive
• Expression II: \(\mathrm{x}^2(\mathrm{x} - \mathrm{y}) = (9)(2) = 18\) ✓ positive
• Expression III: \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y}) = (3)(2)(4) = 24\) ✓ positive

Case 2: \(\mathrm{x} = 3, \mathrm{y} = -1\) (x positive, y negative, \(\mathrm{x} > \mathrm{y}\))

• Expression I: \(\mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y}) = (3)(-1)(3 - (-1)) = (-3)(4) = -12\) ✗ negative
• Expression II: \(\mathrm{x}^2(\mathrm{x} - \mathrm{y}) = (9)(4) = 36\) ✓ positive
• Expression III: \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y}) = (3)(4)(2) = 24\) ✓ positive

Case 3: \(\mathrm{x} = 2, \mathrm{y} = -5\) (x positive, y very negative, \(\mathrm{x} > \mathrm{y}\))

• Expression I: \(\mathrm{x}\mathrm{y}(\mathrm{x} - \mathrm{y}) = (2)(-5)(2 - (-5)) = (-10)(7) = -70\) ✗ negative
• Expression II: \(\mathrm{x}^2(\mathrm{x} - \mathrm{y}) = (4)(7) = 28\) ✓ positive
• Expression III: \(\mathrm{x}(\mathrm{x} - \mathrm{y})(\mathrm{x} + \mathrm{y}) = (2)(7)(-3) = -42\) ✗ negative

Our examples confirm our analysis: only Expression II is always positive!

4. Final Answer

Based on our systematic analysis, only Expression II: \(\mathrm{x}^3 - \mathrm{x}^2\mathrm{y} = \mathrm{x}^2(\mathrm{x} - \mathrm{y})\) must always be positive under the given constraints.

This is because:

• \(\mathrm{x}^2\) is always positive when \(\mathrm{x} > 0\)
• \((\mathrm{x} - \mathrm{y})\) is always positive when \(\mathrm{x} > \mathrm{y}\)
• The product of two positive quantities is always positive

The answer is B. II only.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint about y
Students often assume that since x is positive, y must also be positive. The problem only tells us that \(\mathrm{x} > 0\) and \(\mathrm{x} > \mathrm{y}\), but y could be negative, zero, or positive. This misunderstanding leads students to incorrectly conclude that all expressions will be positive when y is positive.

2. Failing to recognize the need for factoring
Many students try to analyze the expressions in their original form (like \(\mathrm{x}^2\mathrm{y} - \mathrm{x}\mathrm{y}^2\)) rather than factoring them first. Without factoring, it's much harder to see the sign patterns and students may miss the systematic approach of analyzing each factor separately.

3. Not considering all possible cases for y
Even when students realize y can be negative, they often only test one scenario (like y being slightly negative) and miss extreme cases where y could be very negative (like \(\mathrm{y} = -10\) when \(\mathrm{x} = 2\)), which can change the sign of expressions like \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2\).

Errors while executing the approach

1. Sign errors when analyzing factored expressions
When students factor expressions like \(\mathrm{x}\mathrm{y}(\mathrm{x}-\mathrm{y})\), they may correctly identify that \(\mathrm{x} > 0\) and \((\mathrm{x}-\mathrm{y}) > 0\), but then forget to consider that y could be negative, making the overall product \(\mathrm{x}\mathrm{y}(\mathrm{x}-\mathrm{y})\) negative. This is especially common with Expression I.

2. Incorrect factoring of Expression III
Students sometimes factor \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2\) incorrectly. They might stop at \(\mathrm{x}(\mathrm{x}^2 - \mathrm{y}^2)\) and not recognize that \(\mathrm{x}^2 - \mathrm{y}^2\) can be further factored as \((\mathrm{x}-\mathrm{y})(\mathrm{x}+\mathrm{y})\), missing the crucial \((\mathrm{x}+\mathrm{y})\) factor that determines the sign.

3. Arithmetic mistakes in test cases
When testing specific values, students may make calculation errors. For example, with \(\mathrm{x} = 2, \mathrm{y} = -5\), they might incorrectly calculate \((\mathrm{x} + \mathrm{y})\) as positive instead of negative, or make sign errors in the final multiplication.

Errors while selecting the answer

1. Confusing 'sometimes positive' with 'always positive'
Students may find that Expression I or III is positive in their first test case and incorrectly conclude it's always positive, forgetting that the question asks which expressions MUST be positive (meaning positive in ALL valid cases).

2. Misreading the Roman numeral combinations
After correctly determining that only Expression II is always positive, students might accidentally select 'D. I and II' or 'E. II and III' by misreading the answer choices or second-guessing their analysis.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose strategic values for x and y
Since we know \(\mathrm{x} > 0\) and \(\mathrm{x} > \mathrm{y}\), let's test with concrete values that help us explore different scenarios systematically.

Test Case 1: Both x and y positive, with \(\mathrm{x} > \mathrm{y}\)
Let \(\mathrm{x} = 3\) and \(\mathrm{y} = 1\)

I. \(\mathrm{x}^2\mathrm{y} - \mathrm{x}\mathrm{y}^2 = (3)^2(1) - (3)(1)^2 = 9 - 3 = 6\) ✓ (positive)
II. \(\mathrm{x}^3 - \mathrm{x}^2\mathrm{y} = (3)^3 - (3)^2(1) = 27 - 9 = 18\) ✓ (positive)
III. \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2 = (3)^3 - (3)(1)^2 = 27 - 3 = 24\) ✓ (positive)

Test Case 2: x positive, y negative
Let \(\mathrm{x} = 2\) and \(\mathrm{y} = -1\)

I. \(\mathrm{x}^2\mathrm{y} - \mathrm{x}\mathrm{y}^2 = (2)^2(-1) - (2)(-1)^2 = -4 - 2 = -6\) ✗ (negative)
II. \(\mathrm{x}^3 - \mathrm{x}^2\mathrm{y} = (2)^3 - (2)^2(-1) = 8 - (-4) = 8 + 4 = 12\) ✓ (positive)
III. \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2 = (2)^3 - (2)(-1)^2 = 8 - 2 = 6\) ✓ (positive)

Test Case 3: x positive, \(\mathrm{y} = 0\)
Let \(\mathrm{x} = 4\) and \(\mathrm{y} = 0\)

I. \(\mathrm{x}^2\mathrm{y} - \mathrm{x}\mathrm{y}^2 = (4)^2(0) - (4)(0)^2 = 0 - 0 = 0\) ✗ (not positive)
II. \(\mathrm{x}^3 - \mathrm{x}^2\mathrm{y} = (4)^3 - (4)^2(0) = 64 - 0 = 64\) ✓ (positive)
III. \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2 = (4)^3 - (4)(0)^2 = 64 - 0 = 64\) ✓ (positive)

Step 2: Analyze the pattern
From our tests:

• Expression I is sometimes positive, sometimes negative or zero → Not always positive
• Expression II is positive in all test cases → Always positive
• Expression III is positive in all test cases, but we need one more strategic test

Test Case 4: Challenge Expression III with large negative y
Let \(\mathrm{x} = 1\) and \(\mathrm{y} = -2\)

III. \(\mathrm{x}^3 - \mathrm{x}\mathrm{y}^2 = (1)^3 - (1)(-2)^2 = 1 - 4 = -3\) ✗ (negative)

Step 3: Conclusion
Only Expression II is always positive under the given constraints.

Answer: B