If `px^(2) + qx + r = p(x - alpha) (x - beta) and p^(3) + pq + r = 0 : p.q` and r being real number, then which of the following is not possible

Options

A. `alpha = beta = p`
B. `alpha != beta = p`
C. `alpha = beta != p`
D. `beta != alpha = p`

Answer

To solve the problem step by step, we start with the given equation and conditions:

### Step 1: Start with the given equation
We have the equation:
\[ px^2 + qx + r = p(x - \alpha)(x - \beta) \]

### Step 2: Expand the right-hand side
Expanding the right-hand side gives:
\[ p(x - \alpha)(x - \beta) = p(x^2 - (\alpha + \beta)x + \alpha\beta) \]
This simplifies to:
\[ px^2 - p(\alpha + \beta)x + p\alpha\beta \]

### Step 3: Compare coefficients
Now, we can compare the coefficients from both sides:
- Coefficient of \(x^2\): \(p\) (same on both sides)
- Coefficient of \(x\): \(q = -p(\alpha + \beta)\)
- Constant term: \(r = p\alpha\beta\)

### Step 4: Substitute into the second equation
We are given another condition:
\[ p^3 + pq + r = 0 \]
Substituting the expressions for \(q\) and \(r\) from Step 3:
\[ p^3 + p(-p(\alpha + \beta)) + p\alpha\beta = 0 \]
This simplifies to:
\[ p^3 - p^2(\alpha + \beta) + p\alpha\beta = 0 \]

### Step 5: Factor out \(p\)
Factoring out \(p\) gives:
\[ p(p^2 - p(\alpha + \beta) + \alpha\beta) = 0 \]
This implies either:
1. \(p = 0\) (which is not a valid case since it would make the quadratic equation trivial), or
2. \(p^2 - p(\alpha + \beta) + \alpha\beta = 0\)

### Step 6: Solve the quadratic equation
The quadratic equation:
\[ p^2 - p(\alpha + \beta) + \alpha\beta = 0 \]
can be solved using the quadratic formula:
\[ p = \frac{(\alpha + \beta) \pm \sqrt{(\alpha + \beta)^2 - 4\alpha\beta}}{2} \]

### Step 7: Analyze the discriminant
The discriminant of this quadratic is:
\[ D = (\alpha + \beta)^2 - 4\alpha\beta = (\alpha - \beta)^2 \]
Since this is a square, it is always non-negative, which means \(p\) can take real values.

### Conclusion
Since \(p\) can be equal to \(\alpha\) or \(\beta\) based on the roots of the quadratic, we conclude that the conditions provided in the question are satisfied.

### Final Answer
Thus, the question asks which of the following is not possible. Since we derived that \(p\) can equal \(\alpha\) or \(\beta\), we need to check the options provided in the original question to identify the impossible case.