Evaluate : `(2x-y+3z)(4x^(2)+y^(2)+9z^(2)+2xy+3yz-6xz)`

Options

A. `8x^(3)-y^(3)+27z^(3)-18xyz`
B. `8x^(3)-y^(3)+27z^(3)+18xyz`
C. `8x^(3)+y^(3)+27z^(3)+18xyz`
D. `8x^(3)+y^(3)-27z^(3)+18xyz`

Answer

To evaluate the expression \((2x - y + 3z)(4x^2 + y^2 + 9z^2 + 2xy + 3yz - 6xz)\), we can use the identity related to the sum and differences of cubes. The identity states that:

\[ A^3 + B^3 + C^3 - 3ABC = (A + B + C)(A^2 + B^2 + C^2 - AB - BC - CA) \]

### Step-by-Step Solution:

1. **Identify A, B, and C**:
- Let \(A = 2x\), \(B = -y\), and \(C = 3z\).

2. **Rewrite the expression**:
- The polynomial can be rewritten in terms of \(A\), \(B\), and \(C\):
\[ 4x^2 + y^2 + 9z^2 + 2xy + 3yz - 6xz = A^2 + B^2 + C^2 - AB - BC - CA \]

3. **Calculate \(A^2\), \(B^2\), and \(C^2\)**:
- \(A^2 = (2x)^2 = 4x^2\)
- \(B^2 = (-y)^2 = y^2\)
- \(C^2 = (3z)^2 = 9z^2\)

4. **Calculate \(AB\), \(BC\), and \(CA\)**:
- \(AB = (2x)(-y) = -2xy\)
- \(BC = (-y)(3z) = -3yz\)
- \(CA = (3z)(2x) = 6xz\)

5. **Substitute into the identity**:
- Now substituting back into the identity:
\[ A^3 + B^3 + C^3 - 3ABC = (2x - y + 3z)((4x^2 + y^2 + 9z^2) + (2xy + 3yz - 6xz)) \]

6. **Calculate \(A^3\), \(B^3\), and \(C^3\)**:
- \(A^3 = (2x)^3 = 8x^3\)
- \(B^3 = (-y)^3 = -y^3\)
- \(C^3 = (3z)^3 = 27z^3\)

7. **Calculate \(3ABC\)**:
- \(3ABC = 3(2x)(-y)(3z) = -18xyz\)

8. **Combine the results**:
- Therefore, we have:
\[ 8x^3 - y^3 + 27z^3 - (-18xyz) = 8x^3 - y^3 + 27z^3 + 18xyz \]

9. **Final Result**:
- The evaluated expression is:
\[ 8x^3 - y^3 + 27z^3 + 18xyz \]