`x^(4)-x^(2)y^(2)-72y^(4)`=______

Options

A. `(x^(2)+9y^(2))(x^(2)-8y^(2))`
B. `(x+3y)(x+3y)(x^(2)+8y^(2))`
C. `(x+3y)(x-3y)(x^(2)+8y^(2))`
D. `(x^(2)-9y^(2))(x^(2)-8y^(2))`

Answer

To factor the expression \( x^4 - x^2y^2 - 72y^4 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 - x^2y^2 - 72y^4 \]
We can rearrange it to make it easier to factor:
\[ x^4 - 9x^2y^2 + 8x^2y^2 - 72y^4 \]

### Step 2: Group the terms
Now, we can group the terms:
\[ (x^4 - 9x^2y^2) + (8x^2y^2 - 72y^4) \]

### Step 3: Factor out common terms
From the first group \( (x^4 - 9x^2y^2) \), we can factor out \( x^2 \):
\[ x^2(x^2 - 9y^2) \]
From the second group \( (8x^2y^2 - 72y^4) \), we can factor out \( 8y^2 \):
\[ 8y^2(x^2 - 9y^2) \]

### Step 4: Combine the factored terms
Now we can combine the two factored groups:
\[ x^2(x^2 - 9y^2) + 8y^2(x^2 - 9y^2) \]
This gives us:
\[ (x^2 + 8y^2)(x^2 - 9y^2) \]

### Step 5: Factor the difference of squares
The term \( (x^2 - 9y^2) \) can be factored further using the difference of squares:
\[ x^2 - 9y^2 = (x - 3y)(x + 3y) \]

### Final Factored Form
Putting it all together, we have:
\[ (x^2 + 8y^2)(x - 3y)(x + 3y) \]

Thus, the final answer is:
\[ (x^2 + 8y^2)(x - 3y)(x + 3y) \]

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