If p+q+r=11, `pq+qr+rp=8` and pqr=`11/3` then find `p^3+q^3+ r^3` = ?

Options

A. 1056
B. 1078
C. 1038
D. 976

Answer

To find \( p^3 + q^3 + r^3 \) given the equations:

1. \( p + q + r = 11 \)
2. \( pq + qr + rp = 8 \)
3. \( pqr = \frac{11}{3} \)

We can use the identity:

\[ p^3 + q^3 + r^3 = (p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp) + 3pqr \]

### Step 1: Calculate \( p^2 + q^2 + r^2 \)

Using the identity:

\[ p^2 + q^2 + r^2 = (p + q + r)^2 - 2(pq + qr + rp) \]

Substituting the known values:

\[ p^2 + q^2 + r^2 = (11)^2 - 2(8) \]

Calculating:

\[ p^2 + q^2 + r^2 = 121 - 16 = 105 \]

### Step 2: Substitute into the identity for \( p^3 + q^3 + r^3 \)

Now we can substitute \( p^2 + q^2 + r^2 \) back into the equation for \( p^3 + q^3 + r^3 \):

\[ p^3 + q^3 + r^3 = (p + q + r)((p^2 + q^2 + r^2) - (pq + qr + rp)) + 3pqr \]

Substituting the known values:

\[ p^3 + q^3 + r^3 = 11 \left( 105 - 8 \right) + 3 \left( \frac{11}{3} \right) \]

Calculating inside the parentheses:

\[ p^3 + q^3 + r^3 = 11 \times 97 + 11 \]

Calculating the first term:

\[ 11 \times 97 = 1067 \]

Adding the second term:

\[ p^3 + q^3 + r^3 = 1067 + 11 = 1078 \]

### Final Result

Thus, the value of \( p^3 + q^3 + r^3 \) is:

\[ \boxed{1078} \]