If `alpha, beta` are roots of `ax^(2) -2bx +c=0`, then `alpha^(3) beta^(3) + alpha^(2) beta^(3) +alpha^(3) beta^(2)` is

Options

A. `(c^(2)(c+2ab))/(a^(3))`
B. `(bc^(3))/(a^(3))`
C. `(c^(2))/(a^(3))`
D. `none

Answer

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

### Step-by-Step Solution:

1. **Identify the roots and their relationships**:
- From Vieta's formulas, for the quadratic equation \( ax^2 - 2bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = \frac{2b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

2. **Rewrite the expression**:
- The expression we need to evaluate is:
\[ \alpha^3 \beta^3 + \alpha^2 \beta^3 + \alpha^3 \beta^2 \]
- This can be factored as:
\[ \alpha^2 \beta^2 (\alpha + \beta) \]

3. **Substitute the values**:
- We already know:
- \( \alpha + \beta = \frac{2b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)
- Therefore, \( \alpha^2 \beta^2 = (\alpha \beta)^2 = \left(\frac{c}{a}\right)^2 \)

4. **Combine the results**:
- Substitute \( \alpha^2 \beta^2 \) and \( \alpha + \beta \) into the expression:
\[ \alpha^3 \beta^3 + \alpha^2 \beta^3 + \alpha^3 \beta^2 = \left(\frac{c^2}{a^2}\right) \left(\frac{2b}{a}\right) \]

5. **Simplify the expression**:
- This simplifies to:
\[ \frac{2bc^2}{a^3} \]

### Final Result:
Thus, the value of \( \alpha^3 \beta^3 + \alpha^2 \beta^3 + \alpha^3 \beta^2 \) is:
\[ \frac{2bc^2}{a^3} \]