If `a+b+c=0`, then `(a^(3) + b^(3) + c^(3) ) div (abc)` is equal to

Options

A. 1
B. 2
C. 3
D. 9

Answer

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

### Step-by-Step Solution:

1. **Use the Algebraic Identity**:
We start with the algebraic identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
This identity will help us relate \(a^3 + b^3 + c^3\) to \(abc\).

2. **Substitute the Given Condition**:
Since we are given that \(a + b + c = 0\), we can substitute this into the identity:
\[ a^3 + b^3 + c^3 - 3abc = 0 \cdot (a^2 + b^2 + c^2 - ab - ac - bc) \]
The right-hand side simplifies to 0:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]

3. **Rearrange the Equation**:
Now, we can rearrange the equation to isolate \(a^3 + b^3 + c^3\):
\[ a^3 + b^3 + c^3 = 3abc \]

4. **Substitute into the Original Expression**:
We need to find \((a^3 + b^3 + c^3) \div (abc)\):
\[ \frac{a^3 + b^3 + c^3}{abc} = \frac{3abc}{abc} \]

5. **Simplify the Expression**:
Now, we simplify the fraction:
\[ \frac{3abc}{abc} = 3 \]

### Final Answer:
Thus, \((a^3 + b^3 + c^3) \div (abc) = 3\).

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