If `alpha` and `beta` are roots of the equation `px^(2)+qx+1=0`, then the value of `alpha^(3)beta^(2)+alpha^(2)beta^(3)` is

Options

A. `(q)/(p^(3))`
B. `-(q)/(p^(3))`
C. `(p)/(q^(3))`
D. `-(p)/(q^(3))`

Answer

To solve the problem, we need to find the value of \( \alpha^3 \beta^2 + \alpha^2 \beta^3 \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( px^2 + qx + 1 = 0 \).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation:**
The given equation is \( px^2 + qx + 1 = 0 \). Here, we can identify:
- \( a = p \)
- \( b = q \)
- \( c = 1 \)

2. **Use Vieta's formulas to find the sums and products of the roots:**
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{q}{p} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{1}{p} \)

3. **Rewrite the expression \( \alpha^3 \beta^2 + \alpha^2 \beta^3 \):**
We can factor this expression:
\[ \alpha^3 \beta^2 + \alpha^2 \beta^3 = \alpha^2 \beta^2 (\alpha + \beta) \]

4. **Substitute the values of \( \alpha + \beta \) and \( \alpha \beta \):**
From step 2, we have:
- \( \alpha + \beta = -\frac{q}{p} \)
- \( \alpha \beta = \frac{1}{p} \)

Therefore, \( \alpha^2 \beta^2 = (\alpha \beta)^2 = \left(\frac{1}{p}\right)^2 = \frac{1}{p^2} \).

5. **Combine the results:**
Now substituting these values into the factored expression:
\[ \alpha^3 \beta^2 + \alpha^2 \beta^3 = \alpha^2 \beta^2 (\alpha + \beta) = \frac{1}{p^2} \left(-\frac{q}{p}\right) \]

6. **Simplify the expression:**
\[ \frac{1}{p^2} \left(-\frac{q}{p}\right) = -\frac{q}{p^3} \]

### Final Answer:
Thus, the value of \( \alpha^3 \beta^2 + \alpha^2 \beta^3 \) is:
\[ -\frac{q}{p^3} \]