If `x+y+z=0` then what is the value of `(x^2)/(3z)+(y^3)/(3xz)+(z^2)/(3x)`?

Options

A. `0`
B. `xz`
C. `y`
D. `3y`

Answer

To solve the problem, we need to find the value of the expression \(\frac{x^2}{3z} + \frac{y^3}{3xz} + \frac{z^2}{3x}\) given that \(x + y + z = 0\).

### Step-by-Step Solution:

1. **Write the expression**:
We start with the expression:
\[ \frac{x^2}{3z} + \frac{y^3}{3xz} + \frac{z^2}{3x} \]

2. **Find a common denominator**:
The common denominator for the three fractions is \(3xyz\). We will rewrite each term with this common denominator:
\[ \frac{x^2}{3z} = \frac{x^2 \cdot xy}{3xyz} = \frac{x^3y}{3xyz} \]
\[ \frac{y^3}{3xz} = \frac{y^3 \cdot yz}{3xyz} = \frac{y^4z}{3xyz} \]
\[ \frac{z^2}{3x} = \frac{z^2 \cdot zx}{3xyz} = \frac{z^3x}{3xyz} \]

3. **Combine the fractions**:
Now, we can combine the fractions:
\[ \frac{x^3y + y^4z + z^3x}{3xyz} \]

4. **Use the identity for cubes**:
We know from algebra that if \(x + y + z = 0\), then:
\[ x^3 + y^3 + z^3 = 3xyz \]
Thus, we can express \(x^3 + y^3 + z^3\) in terms of \(xyz\).

5. **Substitute the identity**:
We can rewrite the numerator:
\[ x^3 + y^3 + z^3 = 3xyz \]
Therefore, we can replace \(x^3 + y^3 + z^3\) in our expression:
\[ x^3y + y^4z + z^3x = y(x^3 + y^3 + z^3) = y(3xyz) \]

6. **Simplify the expression**:
Substituting this back into our combined fraction gives:
\[ \frac{y(3xyz)}{3xyz} = y \]

7. **Final answer**:
Thus, the value of the expression is:
\[ \boxed{y} \]