The factors of `(x ^(4) - 7x ^(2) y ^(2) + y ^(4))` are

Options

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

Answer

To factor the expression \( x^4 - 7x^2y^2 + y^4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ x^4 - 7x^2y^2 + y^4 \]

2. **Identify a Suitable Form**:
Notice that we can rewrite \(-7x^2y^2\) as \(-9x^2y^2 + 2x^2y^2\):
\[ x^4 + 2x^2y^2 - 9x^2y^2 + y^4 \]

3. **Group the Terms**:
Now, we can group the first three terms and the last term:
\[ (x^4 + 2x^2y^2 + y^4) - 9x^2y^2 \]

4. **Recognize a Perfect Square**:
The expression \(x^4 + 2x^2y^2 + y^4\) can be recognized as a perfect square:
\[ (x^2 + y^2)^2 \]

5. **Substitute Back**:
Substitute back into the expression:
\[ (x^2 + y^2)^2 - 9x^2y^2 \]

6. **Apply the Difference of Squares**:
We can now apply the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\):
Let \(a = x^2 + y^2\) and \(b = 3xy\):
\[ (x^2 + y^2 + 3xy)(x^2 + y^2 - 3xy) \]

7. **Final Factors**:
Thus, the factors of the original expression \(x^4 - 7x^2y^2 + y^4\) are:
\[ (x^2 + y^2 + 3xy)(x^2 + y^2 - 3xy) \]

### Final Answer:
The factors of \(x^4 - 7x^2y^2 + y^4\) are:
\[ (x^2 + y^2 + 3xy)(x^2 + y^2 - 3xy) \]