If `alpha, beta and gamma` are three ral roots of the equatin `x ^(3) -6x ^(2)+5x-1 =0,` then the value of `alpha ^(4) + beta ^(4) + gamma ^(4)` is:

Options

A. 250
B. 650
C. 150
D. 950

Answer

To solve the problem, we need to find the value of \( \alpha^4 + \beta^4 + \gamma^4 \) given that \( \alpha, \beta, \gamma \) are the roots of the cubic equation \( x^3 - 6x^2 + 5x - 1 = 0 \).

### Step 1: Identify the coefficients and apply Vieta's formulas
From the given equation \( x^3 - 6x^2 + 5x - 1 = 0 \), we can identify the coefficients:
- \( a = 1 \)
- \( b = -6 \)
- \( c = 5 \)
- \( d = -1 \)

Using Vieta's formulas, we have:
- \( \alpha + \beta + \gamma = 6 \) (sum of roots)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = 5 \) (sum of products of roots taken two at a time)
- \( \alpha\beta\gamma = 1 \) (product of roots)

### Step 2: Calculate \( \alpha^2 + \beta^2 + \gamma^2 \)
We can find \( \alpha^2 + \beta^2 + \gamma^2 \) using the identity:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Substituting the values we have:
\[ \alpha^2 + \beta^2 + \gamma^2 = 6^2 - 2 \cdot 5 = 36 - 10 = 26 \]

### Step 3: Calculate \( \alpha^4 + \beta^4 + \gamma^4 \)
Next, we use the identity:
\[ \alpha^4 + \beta^4 + \gamma^4 = (\alpha^2 + \beta^2 + \gamma^2)^2 - 2(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2) \]
We already found \( \alpha^2 + \beta^2 + \gamma^2 = 26 \), so we need to calculate \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 \).

### Step 4: Calculate \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 \)
Using the identity:
\[ \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = (\alpha\beta + \beta\gamma + \gamma\alpha)^2 - 2\alpha\beta\gamma(\alpha + \beta + \gamma) \]
Substituting the known values:
\[ \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = 5^2 - 2 \cdot 1 \cdot 6 = 25 - 12 = 13 \]

### Step 5: Substitute back to find \( \alpha^4 + \beta^4 + \gamma^4 \)
Now we can substitute back into the equation for \( \alpha^4 + \beta^4 + \gamma^4 \):
\[ \alpha^4 + \beta^4 + \gamma^4 = 26^2 - 2 \cdot 13 \]
Calculating this gives:
\[ \alpha^4 + \beta^4 + \gamma^4 = 676 - 26 = 650 \]

### Final Answer
Thus, the value of \( \alpha^4 + \beta^4 + \gamma^4 \) is \( \boxed{650} \).