x, y and z are real numbers . If ` x^(3) + y^(3) + z^(3) = 13 , x + y +z= 1 ` and xyz = 1 then what is the value of ` xy + yz + zx ` ?

Options

A. `-1`
B. 1
C. 3
D. `-3`

Answer

To solve the problem step by step, we will use the provided information and the formula for the sum of cubes.

### Given:
1. \( x^3 + y^3 + z^3 = 13 \)
2. \( x + y + z = 1 \)
3. \( xyz = 1 \)

We need to find the value of \( xy + yz + zx \).

### Step 1: Use the formula for the sum of cubes
The formula for the sum of cubes is:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z) \left( (x + y + z)^2 - 3(xy + yz + zx) \right) \]

### Step 2: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 13 - 3(1) = (1) \left( (1)^2 - 3(xy + yz + zx) \right) \]
This simplifies to:
\[ 13 - 3 = 1 \left( 1 - 3(xy + yz + zx) \right) \]
\[ 10 = 1 - 3(xy + yz + zx) \]

### Step 3: Rearranging the equation
Rearranging the equation gives:
\[ 10 = 1 - 3(xy + yz + zx) \]
Subtracting 1 from both sides:
\[ 9 = -3(xy + yz + zx) \]

### Step 4: Solve for \( xy + yz + zx \)
Dividing both sides by -3:
\[ xy + yz + zx = -\frac{9}{3} = -3 \]

### Final Answer
Thus, the value of \( xy + yz + zx \) is:
\[ \boxed{-3} \]