If `x+y+z=6` and `xy+yz +zx =11` then what is the vlaue of `x^(3) +y^(3)+z^(3) -3xyz`?

Options

A. 18
B. 36
C. 54
D. 66

Answer

To find the value of \( x^3 + y^3 + z^3 - 3xyz \) given the equations \( x + y + z = 6 \) and \( xy + yz + zx = 11 \), 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 \( x^2 + y^2 + z^2 \)

We know that:

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

Substituting the known values:

\[ 6^2 = x^2 + y^2 + z^2 + 2(11) \]

Calculating \( 6^2 \):

\[ 36 = x^2 + y^2 + z^2 + 22 \]

Now, isolate \( x^2 + y^2 + z^2 \):

\[ x^2 + y^2 + z^2 = 36 - 22 = 14 \]

### Step 2: Substitute into the identity

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

\[ x^2 + y^2 + z^2 - xy - yz - zx = 14 - 11 = 3 \]

### Step 3: Calculate \( x^3 + y^3 + z^3 - 3xyz \)

Now we substitute back into the identity:

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

Substituting the values we have:

\[ x^3 + y^3 + z^3 - 3xyz = 6 \cdot 3 = 18 \]

### Final Answer

Thus, the value of \( x^3 + y^3 + z^3 - 3xyz \) is \( \boxed{18} \).

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