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

Options

A. 550
B. 525
C. 575
D. 560

Answer

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

1. \( x + y + z = 10 \)
2. \( xy + yz + zx = 15 \)

We can use the identity for the sum of cubes:

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

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

From the first equation, we know:

\[ x + y + z = 10 \]

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

\[ (x + y + z)^2 = 10^2 = 100 \]

### Step 2: Substitute into the identity

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

\[ x^3 + y^3 + z^3 - 3xyz = (10)(100 - 3 \cdot 15) \]

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

Now we calculate \( 3 \cdot 15 \):

\[ 3 \cdot 15 = 45 \]

### Step 4: Substitute back into the equation

Now we substitute this back into our equation:

\[ x^3 + y^3 + z^3 - 3xyz = 10(100 - 45) \]

### Step 5: Simplify the expression

Now we simplify the expression inside the parentheses:

\[ 100 - 45 = 55 \]

So we have:

\[ x^3 + y^3 + z^3 - 3xyz = 10 \cdot 55 \]

### Step 6: Calculate the final result

Now we calculate \( 10 \cdot 55 \):

\[ 10 \cdot 55 = 550 \]

Thus, the final answer is:

\[ \boxed{550} \]