Key Concepts and Formulas

• The equation $x^2 + x + 1 = 0$ has roots $\omega$ and $\omega^2$, where $\omega$ is a non-real cube root of unity.
• Properties of cube roots of unity: $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.
• The identity $1+\omega+\omega^2 = 0$ can be rearranged to $1 + \omega = -\omega^2$, $\omega + \omega^2 = -1$, and $1 + \omega^2 = -\omega$.

Step-by-Step Solution

Step 1: Identify $\alpha$ and Its Properties

Since $\alpha$ satisfies $x^2 + x + 1 = 0$, we can identify $\alpha$ as a non-real cube root of unity, $\omega$. Thus, $\alpha = \omega$, and we have $\alpha^3 = 1$ and $1 + \alpha + \alpha^2 = 0$. This allows us to replace higher powers of $\alpha$ with lower powers, and to express combinations of $1, \alpha, \alpha^2$ in terms of one another.

Step 2: Simplify $(1 + \alpha)^7$

We are given the expression $(1 + \alpha)^7$. Using the property $1 + \alpha + \alpha^2 = 0$, we can write $1 + \alpha = -\alpha^2$. Substituting this into the expression yields:

$(1 + \alpha)^7 = (-\alpha^2)^7 = (-1)^7 (\alpha^2)^7 = -\alpha^{14}$

Now, we simplify $\alpha^{14}$ using the property $\alpha^3 = 1$:

$\alpha^{14} = \alpha^{3 \cdot 4 + 2} = (\alpha^3)^4 \cdot \alpha^2 = (1)^4 \cdot \alpha^2 = \alpha^2$

Therefore,

$(1 + \alpha)^7 = -\alpha^2$

Step 3: Express in the Form $A + B\alpha + C\alpha^2$ with $A, B, C \ge 0$

We have $(1 + \alpha)^7 = -\alpha^2$. We want to express this in the form $A + B\alpha + C\alpha^2$, where $A, B, C \ge 0$. Since $-\alpha^2 = 1 + \alpha$, we can write $-\alpha^2 = 1 + \alpha + 0\alpha^2$ Comparing this expression with $A + B\alpha + C\alpha^2$, we have $A = 1, B = 1, C = 0$. Since $A, B, C \ge 0$, these values satisfy the given condition.

Step 4: Calculate $5(3A - 2B - C)$

Now, substitute the values $A = 1, B = 1, C = 0$ into the expression $5(3A - 2B - C)$:

$5(3A - 2B - C) = 5(3(1) - 2(1) - 0) = 5(3 - 2 - 0) = 5(1) = 5$

Common Mistakes & Tips

• Remember the fundamental properties of cube roots of unity: $\omega^3=1$ and $1+\omega+\omega^2=0$.
• Pay close attention to the signs when substituting $1 + \alpha = -\alpha^2$.
• Ensure that the final values of $A, B, C$ satisfy the non-negativity condition.

Summary

We identified $\alpha$ as a cube root of unity, simplified $(1 + \alpha)^7$ to $-\alpha^2$, expressed it in the form $A + B\alpha + C\alpha^2$ with $A, B, C \ge 0$, and then calculated $5(3A - 2B - C)$. The final result is 5.

Final Answer The final answer is \boxed{5}.