If `a+b+ 2c=0`, then the value of `a^(3) + b^(3) + 8c^(3)` is equal to

Options

A. `3 abc`
B. `4 abc`
C. `abc`
D. `6abc`

Answer

To solve the problem, we need to find the value of \( a^3 + b^3 + 8c^3 \) given that \( a + b + 2c = 0 \).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
We will use the algebraic identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]
Here, we will let \( x = a \), \( y = b \), and \( z = 2c \).

2. **Substituting into the identity**:
Substitute \( x = a \), \( y = b \), and \( z = 2c \) into the identity:
\[ a^3 + b^3 + (2c)^3 - 3ab(2c) = (a + b + 2c)(a^2 + b^2 + (2c)^2 - ab - b(2c) - (2c)a) \]

3. **Calculate \( (2c)^3 \)**:
We know that \( (2c)^3 = 8c^3 \). Thus, we can rewrite the equation as:
\[ a^3 + b^3 + 8c^3 - 6abc = (a + b + 2c)(a^2 + b^2 + 4c^2 - ab - 2bc - 2ac) \]

4. **Using the given condition**:
Since \( a + b + 2c = 0 \), we can substitute this into our equation:
\[ a^3 + b^3 + 8c^3 - 6abc = 0 \cdot (a^2 + b^2 + 4c^2 - ab - 2bc - 2ac) \]
This simplifies to:
\[ a^3 + b^3 + 8c^3 - 6abc = 0 \]

5. **Rearranging the equation**:
Rearranging gives us:
\[ a^3 + b^3 + 8c^3 = 6abc \]

6. **Final Result**:
Therefore, the value of \( a^3 + b^3 + 8c^3 \) is:
\[ \boxed{6abc} \]