The polynomial `x^(3)+8y^(3)+27z^(3)-18xyz` on factorization gives ___________.

Options

A. `(x+2y+3z)(x^(2)+4y^(2)+9z^(2)+2xy+6yz+3zx)`
B. `(x+2y+3z)(x^(2)+4y^(2)+9z^(2)+4xy+12yz+6zx)`
C. `(x+2y+3z)(x^(2)+4y^(2)+9z^(2)+2xy-6yz-3zx)`
D. `(x+2y+3z)(x^(2)+4y^(2)+9z^(2)-2xy-12yz-6zx)`

Answer

To factor the polynomial \( x^3 + 8y^3 + 27z^3 - 18xyz \), we will use the identity for the sum of cubes and the product of three variables.

### Step-by-Step Solution:

1. **Identify the Form**: We recognize that the polynomial can be expressed in the form of \( a^3 + b^3 + c^3 - 3abc \). The identity states:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

2. **Assign Values**: We will assign:
- \( a = x \)
- \( b = 2y \) (since \( 8y^3 = (2y)^3 \))
- \( c = 3z \) (since \( 27z^3 = (3z)^3 \))

3. **Check the Identity**: Now we check if the polynomial fits the identity:
\[ x^3 + (2y)^3 + (3z)^3 - 3 \cdot x \cdot 2y \cdot 3z \]
This simplifies to:
\[ x^3 + 8y^3 + 27z^3 - 18xyz \]
which matches our polynomial.

4. **Apply the Identity**: Now we can apply the identity:
\[ (x + 2y + 3z)(x^2 + (2y)^2 + (3z)^2 - x \cdot 2y - x \cdot 3z - 2y \cdot 3z) \]

5. **Simplify the Second Factor**:
- Calculate \( (2y)^2 = 4y^2 \)
- Calculate \( (3z)^2 = 9z^2 \)
- The second factor becomes:
\[ x^2 + 4y^2 + 9z^2 - 2xy - 3xz - 6yz \]

6. **Final Factorization**: Therefore, the factorization of the polynomial is:
\[ (x + 2y + 3z)(x^2 + 4y^2 + 9z^2 - 2xy - 3xz - 6yz) \]

### Final Answer:
The polynomial \( x^3 + 8y^3 + 27z^3 - 18xyz \) on factorization gives:
\[ (x + 2y + 3z)(x^2 + 4y^2 + 9z^2 - 2xy - 3xz - 6yz) \]