If x+y +z =10, xy +yz +zx =25 and xyz =100 then what is the value of `(x^(3)+y^(3) +z^(3))` ?

Options

A. 450
B. 540
C. 550
D. 570

Answer

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

1. \( x + y + z = 10 \)
2. \( xy + yz + zx = 25 \)
3. \( xyz = 100 \)

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

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

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

We can find \( x^2 + y^2 + z^2 \) using the formula:

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

Substituting the known values:

\[ x^2 + y^2 + z^2 = (10)^2 - 2(25) = 100 - 50 = 50 \]

### Step 2: Substitute into the identity

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

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

Substituting the values we have:

\[ x^3 + y^3 + z^3 = 10 \left(50 - 25\right) + 3 \cdot 100 \]

### Step 3: Simplify the expression

Now we simplify the expression:

\[ x^3 + y^3 + z^3 = 10 \cdot 25 + 300 = 250 + 300 = 550 \]

### Final Answer

Thus, the value of \( x^3 + y^3 + z^3 \) is \( \boxed{550} \).

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