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Algebra
If $a, b, c$ are non-zero real numbers such that $a+b+c = 0$, then what is the value of the expression $\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}$? A) 0 B) 1 C) 3 D) -3
Correct Option: C) 3 To determine the value of the given expression, we start by simplifying it: Given expression: $\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}$ Step 1: Find a common denominator for the terms. The common denominator for $bc$, $ac$, and $ab$ is $abc$. We multiply the numerator and denominator of each fraction by the missing factor to achieve this common denominator: $\frac{a^2 \cdot a}{bc \cdot a} + \frac{b^2 \cdot b}{ac \cdot b} + \frac{c^2 \cdot c}{ab \cdot c} = \frac{a^3}{abc} + \frac{b^3}{abc} + \frac{c^3}{abc}$ Step 2: Combine the fractions. Now, with a common denominator, we can combine the numerators: $\frac{a^3 + b^3 + c^3}{abc}$ Step 3: Apply the given condition using an algebraic identity. We are given that $a+b+c = 0$. A crucial algebraic identity states that for any three numbers $a, b, c$: $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ Since $a+b+c = 0$, the right-hand side of the identity becomes: $0 \cdot (a^2+b^2+c^2-ab-bc-ca) = 0$ Therefore, we have: $a^3+b^3+c^3-3abc = 0$ This implies: $a^3+b^3+c^3 = 3abc$ Step 4: Substitute this result back into the simplified expression. Substitute $3abc$ for $a^3+b^3+c^3$ in our expression from Step 2: $\frac{3abc}{abc}$ Step 5: Simplify the final expression. Since $a, b, c$ are non-zero real numbers, $abc \neq 0$. Thus, we can cancel $abc$ from the numerator and the denominator: $\frac{3abc}{abc} = 3$ Hence, the value of the expression is 3.
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