If `x + y + z = 10, x ^(3) + y ^(3) + z ^(3) = 75 and xyz = 15,` then find the value of `x ^(2) + y ^(2) + z ^(2) - xy - yz - zx`

Options

A. a)3
B. b)5
C. c)6
D. d)4

Answer

To solve the problem, we will use the given equations and a known identity relating the sums of cubes and products of variables.

### Step 1: Write down the given equations
We have:
1. \( x + y + z = 10 \) (Equation 1)
2. \( x^3 + y^3 + z^3 = 75 \) (Equation 2)
3. \( xyz = 15 \) (Equation 3)

### Step 2: Use the identity for the sum of cubes
We can use the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]

### Step 3: Substitute the known values into the identity
Substituting the known values from Equations 1, 2, and 3 into the identity:
\[ 75 - 3 \cdot 15 = 10 (x^2 + y^2 + z^2 - xy - yz - zx) \]

### Step 4: Simplify the left side
Calculating the left side:
\[ 75 - 45 = 30 \]
So we have:
\[ 30 = 10 (x^2 + y^2 + z^2 - xy - yz - zx) \]

### Step 5: Divide both sides by 10
Dividing both sides by 10 gives:
\[ 3 = x^2 + y^2 + z^2 - xy - yz - zx \]

### Final Answer
Thus, the value of \( x^2 + y^2 + z^2 - xy - yz - zx \) is \( \boxed{3} \).

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