If `a+b+c=0` then `((a^2)/(bc)+(b^2)/(ca)+(c^2)/(ab))=?`

Options

A. `1`
B. `0`
C. `-1`
D. `3`

Explanation

To solve the problem, we need to find the value of the expression \(\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}\) given that \(a + b + c = 0\).

### Step-by-Step Solution:

1. **Start with the given condition**:
\[ a + b + c = 0 \]

2. **Rewrite the expression**:
We need to simplify the expression:
\[ \frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} \]

3. **Find a common denominator**:
The common denominator for the fractions is \(abc\). Therefore, we can rewrite the expression as:
\[ \frac{a^2 \cdot a}{abc} + \frac{b^2 \cdot b}{abc} + \frac{c^2 \cdot c}{abc} = \frac{a^3 + b^3 + c^3}{abc} \]

4. **Use the identity for the sum of cubes**:
We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Since \(a + b + c = 0\), it simplifies to:
\[ a^3 + b^3 + c^3 = 3abc \]

5. **Substitute back into the expression**:
Now we can substitute \(a^3 + b^3 + c^3\) into our expression:
\[ \frac{a^3 + b^3 + c^3}{abc} = \frac{3abc}{abc} \]

6. **Simplify the expression**:
This simplifies to:
\[ 3 \]

### Final Answer:
\[ \frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3 \]