The sum of the roots of the equation `5x^2` + (p+q+r) x+ p ar =0 is equal to zero. What is the value of `{p^(3) + q^(3) + r^(3)}`

Options

A. 4pqr
B. 3pqr
C. 7pqr
D. Zpqr

Answer

To solve the problem, we need to determine the value of \( p^3 + q^3 + r^3 \) given that the sum of the roots of the quadratic equation \( 5x^2 + (p + q + r)x + par = 0 \) is equal to zero.

### Step-by-Step Solution:

1. **Understanding the Sum of Roots**:
The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ \text{Sum of roots} = -\frac{b}{a} \]
Here, \( a = 5 \), \( b = (p + q + r) \), and \( c = par \).

2. **Setting Up the Equation**:
According to the problem, the sum of the roots is equal to zero:
\[ -\frac{p + q + r}{5} = 0 \]
This implies:
\[ p + q + r = 0 \]

3. **Using the Identity for Cubes**:
We can use the identity for the sum of cubes:
\[ p^3 + q^3 + r^3 - 3pqr = (p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp) \]
Since we have established that \( p + q + r = 0 \), substituting this into the identity gives:
\[ p^3 + q^3 + r^3 - 3pqr = 0 \cdot (p^2 + q^2 + r^2 - pq - qr - rp) \]
This simplifies to:
\[ p^3 + q^3 + r^3 - 3pqr = 0 \]
Therefore, we can express \( p^3 + q^3 + r^3 \) as:
\[ p^3 + q^3 + r^3 = 3pqr \]

4. **Conclusion**:
The value of \( p^3 + q^3 + r^3 \) is \( 3pqr \).

### Final Answer:
\[ p^3 + q^3 + r^3 = 3pqr \]