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

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the given problem, we start with the equation:

\[ \frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} = 3 \]

We want to find the value of \((x + y + z)^3\).

### Step 1: Multiply through by \(xyz\)

To eliminate the denominators, we multiply both sides of the equation by \(xyz\):

\[ x^2 \cdot x + y^2 \cdot y + z^2 \cdot z = 3xyz \]

This simplifies to:

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

### Step 2: Combine the fractions

Now we can combine the fractions:

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

### Step 3: Clear the fraction

Next, we multiply both sides by \(xyz\):

\[ x^3 + y^3 + z^3 = 3xyz \]

### Step 4: Use the identity for sum of cubes

We can use the identity for the sum of cubes:

\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) \]

From our previous result, we have:

\[ x^3 + y^3 + z^3 - 3xyz = 0 \]

This means:

\[ (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) = 0 \]

### Step 5: Analyze the factors

For the product to be zero, either:

1. \(x + y + z = 0\) or
2. \(x^2 + y^2 + z^2 - xy - xz - yz = 0\)

### Step 6: Evaluate \(x + y + z\)

If \(x + y + z = 0\), then we can substitute this back into our expression for \((x + y + z)^3\):

\[ (x + y + z)^3 = 0^3 = 0 \]

### Final Result

Thus, the value of \((x + y + z)^3\) is:

\[ \boxed{0} \]