Let alpha,beta gamma be the three roots of the equation `x^(3)+bx +c=0`. If `beta gamma=1=-alpha,` then `b^(3)+2c^(3)- 3a(3)-6beta^(3)-8gamma^(3)` is equal to.

Options

A. `19`
B. `21`
C. `(155)/(8)`
D. `(169)/(8)`

Answer

To solve the problem step by step, we start with the given information about the roots of the polynomial equation \(x^3 + bx + c = 0\).

### Step 1: Understand the roots
Let \(\alpha, \beta, \gamma\) be the roots of the equation. We are given that:
\[ \beta \gamma = 1 \quad \text{and} \quad \alpha = -1. \]

### Step 2: Use Vieta's Formulas
From Vieta's formulas, we know:
1. The sum of the roots:
\[ \alpha + \beta + \gamma = 0 \implies -1 + \beta + \gamma = 0 \implies \beta + \gamma = 1. \]
2. The sum of the products of the roots taken two at a time:
\[ \alpha \beta + \beta \gamma + \alpha \gamma = b. \]

### Step 3: Substitute \(\alpha\) into the equations
Substituting \(\alpha = -1\) into the second equation:
\[ -1 \cdot \beta + \beta \cdot \gamma - 1 \cdot \gamma = b \implies -\beta + 1 - \gamma = b. \]
Since \(\beta + \gamma = 1\), we can express \(\gamma\) as:
\[ \gamma = 1 - \beta. \]
Substituting this into the equation gives:
\[ -\beta + 1 - (1 - \beta) = b \implies -\beta + 1 - 1 + \beta = b \implies b = 0. \]

### Step 4: Find \(c\)
Using the product of the roots:
\[ \alpha \beta \gamma = -c \implies (-1) \cdot \beta \cdot \gamma = -c. \]
Since \(\beta \gamma = 1\), we have:
\[ -1 = -c \implies c = 1. \]

### Step 5: Substitute \(b\) and \(c\) into the expression
Now we substitute \(b = 0\) and \(c = 1\) into the expression:
\[ b^3 + 2c^3 - 3\alpha^3 - 6\beta^3 - 8\gamma^3. \]
Calculating each term:
1. \(b^3 = 0^3 = 0\),
2. \(c^3 = 1^3 = 1 \implies 2c^3 = 2 \cdot 1 = 2\),
3. \(\alpha^3 = (-1)^3 = -1 \implies -3\alpha^3 = -3(-1) = 3\).

### Step 6: Calculate \(\beta^3\) and \(\gamma^3\)
Since \(\beta\) and \(\gamma\) are the roots of the quadratic \(x^2 - x + 1 = 0\), we can find:
- The roots are \(\beta = -\omega\) and \(\gamma = -\omega^2\) where \(\omega = e^{2\pi i / 3}\) (the cube roots of unity).
- We know \(\beta^3 = (-\omega)^3 = -\omega^3 = -1\) and \(\gamma^3 = (-\omega^2)^3 = -(\omega^2)^3 = -1\).

Thus:
\[ -6\beta^3 = -6(-1) = 6, \]
\[ -8\gamma^3 = -8(-1) = 8. \]

### Step 7: Combine all the terms
Now we combine all the terms:
\[ 0 + 2 + 3 + 6 + 8 = 19. \]

### Final Answer
Thus, the value of the expression \(b^3 + 2c^3 - 3\alpha^3 - 6\beta^3 - 8\gamma^3\) is:
\[ \boxed{19}. \]