factors of `x^(4)+y^(4)-x^(2)y^(2)` = ___________.

Options

A. `(x^(2)+y^(2)+sqrt(3)xy)(x^(2)+y^(2)-sqrt(3)xy)`
B. `(x^(2)-y^(2)+sqrt(3)xy)(x^(2)-y^(2)-sqrt(3)xy)`
C. `(x^(2)-y^(2)+sqrt(3)xy)(y^(2)-x^(2)-sqrt(3)xy)`
D. `(x^(2)+y^(2)-sqrt(3)xy)(x^(2)+y^(2)-sqrt(3)xy)`

Answer

To factor the expression \( x^4 + y^4 - x^2y^2 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 + y^4 - x^2y^2 \]

### Step 2: Recognize a pattern
Notice that \( x^4 + y^4 \) can be rewritten using the identity for the square of a sum:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 \]
Thus, we can rewrite the expression as:
\[ (x^2 + y^2)^2 - 2x^2y^2 - x^2y^2 \]
This simplifies to:
\[ (x^2 + y^2)^2 - 3x^2y^2 \]

### Step 3: Use the difference of squares
Now we have a difference of squares:
\[ (x^2 + y^2)^2 - (\sqrt{3}xy)^2 \]
We can apply the difference of squares formula, which states that \( a^2 - b^2 = (a + b)(a - b) \). Here, let \( a = x^2 + y^2 \) and \( b = \sqrt{3}xy \):
\[ (x^2 + y^2 + \sqrt{3}xy)(x^2 + y^2 - \sqrt{3}xy) \]

### Step 4: Write the final factors
Thus, the factors of the expression \( x^4 + y^4 - x^2y^2 \) are:
\[ (x^2 + y^2 + \sqrt{3}xy)(x^2 + y^2 - \sqrt{3}xy) \]

### Final Answer
The factors of \( x^4 + y^4 - x^2y^2 \) are:
\[ (x^2 + y^2 + \sqrt{3}xy)(x^2 + y^2 - \sqrt{3}xy) \]

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