`{:("Column A", y > 0,"Column B"),(y^3 + y^4," ", y^4 - 2y^2):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the expressions in Column A and Column B given that \( y > 0 \).

### Step 1: Write down the expressions for Column A and Column B
- Column A: \( y^3 + y^4 \)
- Column B: \( y^4 - 2y^2 \)

### Step 2: Analyze the expressions
Since \( y > 0 \), we know that:
- \( y^3 > 0 \)
- \( y^4 > 0 \)
- \( y^2 > 0 \)

This means all terms in both columns are positive.

### Step 3: Simplify Column A
Column A can be expressed as:
\[ y^3 + y^4 = y^4 + y^3 \]
This shows that Column A is the sum of two positive terms.

### Step 4: Simplify Column B
Column B can be expressed as:
\[ y^4 - 2y^2 \]
Here, \( 2y^2 \) is also positive since \( y^2 > 0 \). Therefore, we can say:
\[ y^4 - 2y^2 < y^4 \]
This indicates that Column B is less than \( y^4 \).

### Step 5: Compare the two columns
Now we compare the two columns:
- Column A: \( y^4 + y^3 \) (which is greater than \( y^4 \))
- Column B: \( y^4 - 2y^2 \) (which is less than \( y^4 \))

Since Column A is greater than \( y^4 \) and Column B is less than \( y^4 \), it follows that:
\[ \text{Column A} > \text{Column B} \]

### Conclusion
Thus, we conclude that Column A is always greater than Column B when \( y > 0 \).

### Final Answer
Column A is greater than Column B.

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