One of the factors of `a^(6)+b^(6)-a^(2)b^(4)-a^(4)b^(2)` is

Options

A. `a+1`
B. `a^(2)+b^(2)`
C. `a^(3)+b^(3)`
D. `a^(4)+b^(4)`

Answer

To find one of the factors of the expression \( a^6 + b^6 - a^2b^4 - a^4b^2 \), we can follow these steps:

### Step 1: Rewrite the expression
The given expression is:
\[ a^6 + b^6 - a^2b^4 - a^4b^2 \]

### Step 2: Group the terms
We can group the terms in pairs:
\[ (a^6 - a^4b^2) + (b^6 - a^2b^4) \]

### Step 3: Factor out common terms
From the first group \( a^6 - a^4b^2 \), we can factor out \( a^4 \):
\[ a^4(a^2 - b^2) \]
From the second group \( b^6 - a^2b^4 \), we can factor out \( b^4 \):
\[ b^4(b^2 - a^2) \]
So, we have:
\[ a^4(a^2 - b^2) + b^4(b^2 - a^2) \]

### Step 4: Notice the common factor
Now, we can see that \( b^2 - a^2 = -(a^2 - b^2) \). Thus, we can write:
\[ a^4(a^2 - b^2) - b^4(a^2 - b^2) \]

### Step 5: Factor out \( (a^2 - b^2) \)
Now we can factor out \( (a^2 - b^2) \):
\[ (a^2 - b^2)(a^4 - b^4) \]

### Step 6: Factor \( a^4 - b^4 \)
The expression \( a^4 - b^4 \) can be factored further using the difference of squares:
\[ a^4 - b^4 = (a^2 + b^2)(a^2 - b^2) \]

### Step 7: Combine the factors
Now we can combine all the factors:
\[ (a^2 - b^2)(a^2 + b^2)(a^2 - b^2) \]
This simplifies to:
\[ (a^2 - b^2)^2(a^2 + b^2) \]

### Final Result
Thus, one of the factors of the expression \( a^6 + b^6 - a^2b^4 - a^4b^2 \) is:
\[ a^2 - b^2 \]

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