If the sum of the two roots of `x^3 + px^2 + ax + r = 0` is zero then `pq=`

Options

A. `-r`
B. r
C. 2r
D. `-2r`

Explanation

To solve the problem, we need to find the value of \( pq \) given that the sum of two roots of the cubic equation \( x^3 + px^2 + ax + r = 0 \) is zero.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the cubic equation be \( \alpha, \beta, \) and \( \gamma \). According to the problem, the sum of two of the roots is zero. We can assume:
\[ \alpha + \beta = 0 \]

2. **Express the Third Root**:
From the equation above, we can express \( \beta \) in terms of \( \alpha \):
\[ \beta = -\alpha \]
Now, using Vieta's formulas, we know that the sum of the roots \( \alpha + \beta + \gamma \) is equal to \( -p \):
\[ \alpha + (-\alpha) + \gamma = -p \]
This simplifies to:
\[ \gamma = -p \]

3. **Substitute the Third Root into the Cubic Equation**:
Since \( \gamma \) is a root of the equation, we can substitute \( \gamma = -p \) into the cubic equation:
\[ (-p)^3 + p(-p)^2 + a(-p) + r = 0 \]

4. **Simplify the Equation**:
Now, we simplify the left-hand side:
\[ -p^3 + p(p^2) - ap + r = 0 \]
This simplifies to:
\[ -p^3 + p^3 - ap + r = 0 \]
The \( -p^3 \) and \( p^3 \) cancel each other out:
\[ -ap + r = 0 \]

5. **Rearrange to Find \( r \)**:
Rearranging gives us:
\[ r = ap \]

6. **Find \( pq \)**:
We need to find \( pq \). From the previous step, we have \( r = ap \). To find \( pq \), we can express \( q \) in terms of \( r \) and \( p \):
\[ pq = p \cdot q \]
Since \( r = ap \), we can express \( q \) as:
\[ q = \frac{r}{p} \]
Thus:
\[ pq = p \cdot \frac{r}{p} = r \]

### Final Answer:
Thus, the value of \( pq \) is equal to \( r \).