If ` (x + y + z) = 12, xy + yz + zx = 44 " and " xyz = 48`, then what is the value of `x^(3) + y^(3) + z^(3) `?

Options

A. `104`
B. `144`
C. `196`
D. `288`

Answer

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

1. \( x + y + z = 12 \)
2. \( xy + yz + zx = 44 \)
3. \( xyz = 48 \)

We can use the identity that relates the sum of cubes to the sums and products of the variables:

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

### Step 1: Calculate \( (x + y + z)^2 \)

Using the first equation:

\[ (x + y + z)^2 = 12^2 = 144 \]

### Step 2: Substitute into the identity

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

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

Substituting the values we have:

\[ x^3 + y^3 + z^3 - 3(48) = 12 \left( 144 - 3(44) \right) \]

### Step 3: Calculate \( 3(xy + yz + zx) \)

Calculating \( 3(44) \):

\[ 3(44) = 132 \]

### Step 4: Substitute and simplify

Now substitute this back into the equation:

\[ x^3 + y^3 + z^3 - 144 = 12(144 - 132) \]

Calculating \( 144 - 132 \):

\[ 144 - 132 = 12 \]

Now substitute this back into the equation:

\[ x^3 + y^3 + z^3 - 144 = 12 \times 12 \]

Calculating \( 12 \times 12 \):

\[ 12 \times 12 = 144 \]

### Step 5: Solve for \( x^3 + y^3 + z^3 \)

Now we have:

\[ x^3 + y^3 + z^3 - 144 = 144 \]

Adding 144 to both sides:

\[ x^3 + y^3 + z^3 = 144 + 144 = 288 \]

Thus, the value of \( x^3 + y^3 + z^3 \) is \( 288 \).

### Final Answer

\[ \boxed{288} \]