Factorize `x^4y^4-xy`

Explanation

To factorize the expression \( x^4y^4 - xy \), we can follow these steps:

### Step 1: Identify Common Factors
First, we notice that both terms in the expression \( x^4y^4 \) and \( -xy \) have a common factor of \( xy \).

### Step 2: Factor Out the Common Factor
We can factor out \( xy \) from the expression:
\[ x^4y^4 - xy = xy(x^3y^3 - 1) \]

### Step 3: Recognize the Difference of Cubes
Now, we observe that \( x^3y^3 - 1 \) is a difference of cubes. We can use the identity for the difference of cubes, which states:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, we can let \( a = xy \) and \( b = 1 \).

### Step 4: Apply the Difference of Cubes Identity
Using the identity, we can rewrite \( x^3y^3 - 1 \):
\[ x^3y^3 - 1 = (xy - 1)((xy)^2 + (xy)(1) + 1^2) \]
This simplifies to:
\[ = (xy - 1)(x^2y^2 + xy + 1) \]

### Step 5: Combine the Factors
Now, substituting back into our expression, we have:
\[ xy(x^3y^3 - 1) = xy(xy - 1)(x^2y^2 + xy + 1) \]

### Final Factored Form
Thus, the complete factorization of \( x^4y^4 - xy \) is:
\[ \boxed{xy(xy - 1)(x^2y^2 + xy + 1)} \]

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