What will be factors of `(a^(2)-b^(2))^(3) + (b^(2) -c^(2))^(3) + (c^(2) - a^(2))^(3)`

Options

A. `(a+b) (a-b)`
B. `(a+b) (a+b)`
C. `(a-c) (a-c)`
D. `(b-c) (b-c)`

Answer

To find the factors of the expression \((a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3\), we can use the identity for the sum of cubes. Here’s a step-by-step solution:

### Step 1: Identify the terms
Let:
- \(x = a^2 - b^2\)
- \(y = b^2 - c^2\)
- \(z = c^2 - a^2\)

We need to analyze the expression \(x^3 + y^3 + z^3\).

### Step 2: Use the identity for the sum of cubes
The identity states that if \(x + y + z = 0\), then:
\[ x^3 + y^3 + z^3 = 3xyz \]
We will check if \(x + y + z = 0\).

### Step 3: Calculate \(x + y + z\)
\[ x + y + z = (a^2 - b^2) + (b^2 - c^2) + (c^2 - a^2) \]
Simplifying this:
\[ = a^2 - b^2 + b^2 - c^2 + c^2 - a^2 = 0 \]

### Step 4: Apply the identity
Since \(x + y + z = 0\), we can apply the identity:
\[ x^3 + y^3 + z^3 = 3xyz \]
Thus,
\[ (a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 = 3(a^2 - b^2)(b^2 - c^2)(c^2 - a^2) \]

### Step 5: Factor each term
Next, we can factor each of the quadratic differences:
\[ a^2 - b^2 = (a - b)(a + b) \]
\[ b^2 - c^2 = (b - c)(b + c) \]
\[ c^2 - a^2 = (c - a)(c + a) \]

### Step 6: Combine the factors
Putting it all together, we have:
\[ 3(a^2 - b^2)(b^2 - c^2)(c^2 - a^2) = 3[(a - b)(a + b)][(b - c)(b + c)][(c - a)(c + a)] \]

### Final Result
Thus, the factors of the original expression are:
\[ 3(a - b)(a + b)(b - c)(b + c)(c - a)(c + a) \]