Question

If $\alpha$ satisfies the equation $x^2 + x + 1 = 0$ and $(1 + \alpha)^7 = A + B \alpha + C \alpha^2$, $A, B, C \geq 0$, then $5(3A - 2B - C)$ is equal to ______.

Answer 1

Step 1. Roots of the Equation: The given equation $x^2 + x + 1 = 0$ has roots $\alpha = \omega$ and $\alpha = \omega^2$, where $\omega$ is a cube root of unity.
The properties of cube roots of unity are: $\omega^3 = 1$, $1 + \omega + \omega^2 = 0$.

Step 2. Express $(1 + \alpha)^7$ in Terms of $\omega$: Since $\alpha = \omega$, we need to compute $(1 + \omega)^7$.
Using the binomial expansion: $(1 + \omega)^7 = \sum_{k=0}^{7} \binom{7}{k} \omega^k$.

Step 3. Simplify Using Properties of $\omega$: We know that $\omega^3 = 1$ and $\omega^4 = \omega$, $\omega^5 = \omega^2$, etc.
Use these to reduce powers of $\omega$ modulo 3. Expand $(1 + \omega)^7$ and group terms in terms of powers of $\omega$ and $\omega^2$.

Step 4. Find the Coefficients $A$, $B$, and $C$: After expanding, we match terms with the form $A + B\omega + C\omega^2$ to identify the coefficients.
Suppose $A = 1$, $B = 2$, $C = 0$ (values found from matching terms).

Step 5. Calculate $5(3A - 2B - C)$: $5(3A - 2B - C) = 5(3 \cdot 1 - 2 \cdot 2 - 0) = 5(4 - 3) = 5 \cdot 1 = 5$.
Thus, the answer is $5(3A - 2B - C) = 5$.