What are the factors of `(a^(3)-(sqrt(2b))^(3))` ?

Options

A. `(a-sqrt(2)b)(a^(2)+sqrt(2)ab+2b)`
B. `(a-sqrt(2)b)(a^(2)+sqrt(2)ab+2b^(2))`
C. `(a-sqrt(2)b)(a^(2)+sqrt(2)ab+b^(2))`
D. None of above

Answer

To factor the expression \( a^3 - (\sqrt{2b})^3 \), we can use the formula for the difference of cubes, which states:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

### Step-by-Step Solution:

1. **Identify \( x \) and \( y \)**:
- Here, we can let \( x = a \) and \( y = \sqrt{2b} \).

2. **Apply the difference of cubes formula**:
- According to the formula, we can write:
\[ a^3 - (\sqrt{2b})^3 = (a - \sqrt{2b})(a^2 + a(\sqrt{2b}) + (\sqrt{2b})^2) \]

3. **Simplify the second factor**:
- Now, we need to simplify \( a^2 + a(\sqrt{2b}) + (\sqrt{2b})^2 \):
- The term \( (\sqrt{2b})^2 = 2b \).
- Thus, we have:
\[ a^2 + a(\sqrt{2b}) + 2b \]

4. **Combine the factors**:
- Therefore, the complete factorization of \( a^3 - (\sqrt{2b})^3 \) is:
\[ (a - \sqrt{2b})(a^2 + a\sqrt{2b} + 2b) \]

### Final Answer:
The factors of \( a^3 - (\sqrt{2b})^3 \) are:
\[ (a - \sqrt{2b})(a^2 + a\sqrt{2b} + 2b) \]