Key Concepts and Formulas

• Vieta's Formulas: For a cubic equation $x^3 + bx + c = 0$ with roots $\alpha, \beta, \gamma$:
  • $\alpha + \beta + \gamma = 0$
  • $\alpha\beta + \beta\gamma + \gamma\alpha = b$
  • $\alpha\beta\gamma = -c$
• Roots of Unity: The solutions to $x^3 = -1$ are $-1, -\omega, -\omega^2$, where $\omega = e^{i2\pi/3}$ is a complex cube root of unity. Key properties include $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.

Step-by-Step Solution

Step 1: Determine the value of $c$ using the product of roots.

We are given that $\alpha, \beta, \gamma$ are the roots of $x^3 + bx + c = 0$. From Vieta's formulas, the product of the roots is $\alpha\beta\gamma = -c$. We are also given that $\beta\gamma = 1$ and $\alpha = -1$.

• Reasoning: The product of the roots formula relates the roots to the constant term $c$. We can use the given conditions for $\alpha, \beta, \gamma$ to find the value of $c$.

Substituting $\alpha = -1$ and $\beta\gamma = 1$ into the product of roots formula: $(-1)(1) = -c$ $-1 = -c$ Therefore, $c = 1$

Step 2: Determine the value of $b$ using the fact that $\alpha$ is a root.

Since $\alpha = -1$ is a root of the equation $x^3 + bx + c = 0$, it must satisfy the equation when substituted for $x$.

• Reasoning: If a value is a root of an equation, substituting it into the equation will result in the equation being equal to zero. This allows us to find the coefficient $b$.

Substituting $x = -1$ into the equation: $(-1)^3 + b(-1) + c = 0$ $-1 - b + c = 0$ Substituting the value of $c = 1$ that we found in Step 1: $-1 - b + 1 = 0$ $-b = 0$ Therefore, $b = 0$

Step 3: Identify the cubic equation and its roots.

Now that we have $b = 0$ and $c = 1$, the original cubic equation $x^3 + bx + c = 0$ becomes: $x^3 + (0)x + 1 = 0$ $x^3 + 1 = 0$ This can be rewritten as $x^3 = -1$.

• Reasoning: By finding the coefficients $b$ and $c$, we have determined the specific cubic equation. Recognizing this equation as $x^3 = -1$ is crucial because its roots are standard complex numbers related to the cube roots of unity.

The roots of $x^3 = -1$ are $-1, -\omega, -\omega^2$, where $\omega$ is a complex cube root of unity. We already know that one root is $\alpha = -1$. Therefore, the other two roots, $\beta$ and $\gamma$, must be $-\omega$ and $-\omega^2$. Let's verify this assignment with the given condition $\beta\gamma = 1$: If we assign $\beta = -\omega$ and $\gamma = -\omega^2$: $\beta\gamma = (-\omega)(-\omega^2) = \omega^3$ Since $\omega^3 = 1$, we have: $\beta\gamma = 1$ This is consistent with the given condition. So, we have:

• $\alpha = -1$
• $\beta = -\omega$
• $\gamma = -\omega^2$

Step 4: Evaluate the given expression.

We need to calculate the value of the expression: $b^3 + 2c^3 - 3\alpha^3 - 6\beta^3 - 8\gamma^3$

• Reasoning: With all the roots and coefficients determined, we can now substitute these values into the target expression. The properties of $\omega$ will be essential for simplifying terms like $(-\omega)^3$ and $(-\omega^2)^3$.

Substituting the values: $b = 0$, $c = 1$, $\alpha = -1$, $\beta = -\omega$, $\gamma = -\omega^2$: $(0)^3 + 2(1)^3 - 3(-1)^3 - 6(-\omega)^3 - 8(-\omega^2)^3$ Simplifying each term:

• $0^3 = 0$
• $2(1)^3 = 2(1) = 2$
• $3(-1)^3 = 3(-1) = -3$
• $6(-\omega)^3 = 6(-1)^3(\omega^3) = 6(-1)(1) = -6$
• $8(-\omega^2)^3 = 8(-1)^3(\omega^6) = 8(-1)(\omega^3)^2 = 8(-1)(1)^2 = -8$

Now, substitute these simplified terms back into the expression: $0 + 2 - (-3) - (-6) - (-8)$ $0 + 2 + 3 + 6 + 8$ $19$

Common Mistakes & Tips

• Sign Errors in Vieta's Formulas: Ensure correct signs. The product of roots is $-D/A$ for $Ax^3+Bx^2+Cx+D=0$.
• Incorrect Identification of Cube Roots of Unity: Remember the roots of $x^3 = -1$ are $-1, -\omega, -\omega^2$.
• Powers of $\omega$: Use $\omega^3 = 1$ to simplify higher powers.

Summary

We used Vieta's formulas and the properties of cube roots of unity to solve this problem. We first determined the coefficients $b$ and $c$ of the cubic equation. Then, using the fact that $x^3 = -1$, we identified the roots $\alpha, \beta, \gamma$. Finally, we substituted these values into the given expression and simplified to find the result.

The final answer is \boxed{19}, which corresponds to option (B).