If \( x + y + z = 0 \), what is the value of the expression: \[ \frac{x^2}{yz} + \frac{y^2}{xz} + \frac{z^2}{xy}? \]

Question: If x+y+z=0, what is the value of the expression:

$$\frac{x^2}{yz}+\frac{y^2}{xz}+\frac{z^2}{xy}?$$

Answer: 

We need to evaluate the expression:

$$\frac{x^2}{yz} + \frac{y^2}{xz} + \frac{z^2}{xy}$$

given that:

$$x + y + z = 0$$

Step 1: Rewrite the Expression

Rewriting each term:

$$\frac{x^2}{yz} + \frac{y^2}{xz} + \frac{z^2}{xy} = \frac{x^3 + y^3 + z^3}{xyz}$$

Using the algebraic identity:

$$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$

Since we are given that $x + y + z = 0$, the term $(x + y + z)$ becomes 0, making:

$$x^3 + y^3 + z^3 = 3xyz$$

Step 2: Substitute in the Expression

$$\frac{x^3 + y^3 + z^3}{xyz} = \frac{3xyz}{xyz} = 3$$

Final Answer:

$$\boxed{{\displaystyle 3}}$$

Additional Information :

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