If $a + b + c = 0$, the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$ is

### Concept & Formula This tests the fundamental condition of cubic identities: If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$. ### Step-by-Step Solution * **Given:** $a + b + c = 0$. * Write out the expression to be evaluated: $$\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$$ * Find a common denominator for the three fractions, which is $abc$. * Multiply the numerator and denominator of each fraction to achieve the common denominator: $$= \frac{a^2 \cdot a}{abc} + \frac{b^2 \cdot b}{abc} + \frac{c^2 \cdot c}{abc}$$ * Combine into a single fraction: $$= \frac{a^3 + b^3 + c^3}{abc}$$ * Apply the conditional identity (since $a + b + c = 0$): $$a^3 + b^3 + c^3 = 3abc$$ * Substitute this back into the numerator: $$= \frac{3abc}{abc}$$ * Cancel $abc$ from the top and bottom (assuming $a, b, c \neq 0$): $$= 3$$ ### Exam Strategy & Shortcut You can verify this quickly by picking small integers that sum to zero. For example, let $a=1, b=1, c=-2$. Substitute: $1/(-2) + 1/(-2) + 4/(1) = -0.5 - 0.5 + 4 = 3$. This confirms the result instantly without algebra. ### Common Pitfall Forgetting to find a common denominator and attempting to cancel terms prematurely, or mistakenly assuming $a^3 + b^3 + c^3 = 0$. ### Final Answer Therefore, the correct answer is **3**.