A factor of `a^(4) -11 a^(2) b^(2) +b^(4)` is

Options

A. `(a^(2) -b^(2) -3ab)`
B. `a^(2) +b^(2) -3ab`
C. `(a^(2) +b^(2) +3ab)`
D. `(a^(2) -b^(2) +4ab)`

Answer

To find a factor of the expression \( a^4 - 11a^2b^2 + b^4 \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ a^4 - 11a^2b^2 + b^4 \]

### Step 2: Rearrange the Middle Term
We can rewrite the middle term \(-11a^2b^2\) as a combination of two terms:
\[ a^4 - 2a^2b^2 - 9a^2b^2 + b^4 \]

### Step 3: Group the Terms
Now, we group the terms:
\[ (a^4 - 2a^2b^2 + b^4) - 9a^2b^2 \]

### Step 4: Recognize a Perfect Square
The first part \(a^4 - 2a^2b^2 + b^4\) can be recognized as a perfect square:
\[ (a^2 - b^2)^2 \]

### Step 5: Substitute Back
Now substituting back, we have:
\[ (a^2 - b^2)^2 - 9a^2b^2 \]

### Step 6: Apply the Difference of Squares
This expression can be factored using the difference of squares:
\[ [(a^2 - b^2) - 3ab][(a^2 - b^2) + 3ab] \]

### Step 7: Factor Further
Now, we can factor \(a^2 - b^2\) further:
\[ (a - b)(a + b) - 3ab \quad \text{and} \quad (a - b)(a + b) + 3ab \]

### Step 8: Final Factors
Thus, the complete factorization of the original expression is:
\[ (a^2 - b^2 - 3ab)(a^2 - b^2 + 3ab) \]

### Conclusion
The factors of the expression \( a^4 - 11a^2b^2 + b^4 \) are:
\[ (a^2 - b^2 - 3ab) \quad \text{and} \quad (a^2 - b^2 + 3ab) \]