If `alpha, beta, gamma` are the roiots of `x^(3) - 10x^(2) +6x - 8 = 0 ` then `alpha^(2) + beta^(2) + gamma^(2)` =

Options

A. `-88`
B. `88`
C. `-7`
D. 1

Answer

To find the value of \( \alpha^2 + \beta^2 + \gamma^2 \) where \( \alpha, \beta, \gamma \) are the roots of the polynomial equation \( x^3 - 10x^2 + 6x - 8 = 0 \), we can use the relationships provided by Vieta's formulas.

### Step-by-step Solution:

1. **Identify the coefficients**:
The given polynomial is \( x^3 - 10x^2 + 6x - 8 = 0 \).
- Here, \( A = 1 \), \( B = -10 \), \( C = 6 \), and \( D = -8 \).

2. **Use Vieta's formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{B}{A} = -\frac{-10}{1} = 10 \).
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{C}{A} = \frac{6}{1} = 6 \).

3. **Use the formula for the sum of squares**:
We know the formula for the sum of squares of the roots:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]

4. **Substitute the values**:
- Substitute \( \alpha + \beta + \gamma = 10 \) and \( \alpha\beta + \beta\gamma + \gamma\alpha = 6 \):
\[ \alpha^2 + \beta^2 + \gamma^2 = (10)^2 - 2(6) \]

5. **Calculate**:
- Calculate \( (10)^2 = 100 \).
- Calculate \( 2(6) = 12 \).
- Now, substitute these values:
\[ \alpha^2 + \beta^2 + \gamma^2 = 100 - 12 = 88 \]

### Final Answer:
\[ \alpha^2 + \beta^2 + \gamma^2 = 88 \]