If `x + y + z = 19, x^(2) + y^(2) + z^(2) = 133`, then the value of `x^(3) + y^(3) + z^(3) - 3xyz` is :

Options

A. 361
B. 342
C. 380
D. 352

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + z^3 - 3xyz \) given the equations:

1. \( x + y + z = 19 \)
2. \( x^2 + y^2 + z^2 = 133 \)

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 1: Calculate \( xy + yz + zx \)

From the identity:

\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]

We can rearrange this to find \( xy + yz + zx \):

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

Substituting the values we have:

\[ xy + yz + zx = \frac{19^2 - 133}{2} \]

Calculating \( 19^2 \):

\[ 19^2 = 361 \]

Now substituting back:

\[ xy + yz + zx = \frac{361 - 133}{2} = \frac{228}{2} = 114 \]

### Step 2: Substitute into the identity

Now we can substitute \( x + y + z \), \( x^2 + y^2 + z^2 \), and \( xy + yz + zx \) into the identity:

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

Substituting the known values:

\[ x^3 + y^3 + z^3 - 3xyz = 19 \left( 133 - 114 \right) \]

Calculating \( 133 - 114 \):

\[ 133 - 114 = 19 \]

Now substituting this back into the equation:

\[ x^3 + y^3 + z^3 - 3xyz = 19 \times 19 = 361 \]

### Final Answer

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

\[ \boxed{361} \]