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

Correct Answer: 5

Given 

Let \( \alpha \) satisfy \( \alpha^2+\alpha+1=0 \). Then the roots are cube roots of unity (other than 1): \( \alpha=\omega \) or \( \omega^2 \) with \( \omega^3=1,\ \omega\neq1 \).

Goal

Express \( (1+\alpha)^7 \) in the form \( A+B\alpha+C\alpha^2 \) and compute \[ 5(3A-2B-C). \]

Solution

Choose \( \alpha=\omega \). Using \( 1+\omega+\omega^2=0 \Rightarrow 1+\omega=-\omega^2 \): \[ (1+\alpha)^7=(1+\omega)^7=(-\omega^2)^7=-\omega^{14}. \] Since \( \omega^3=1 \Rightarrow \omega^{12}=1 \), we have \( \omega^{14}=\omega^2 \). Hence \[ (1+\alpha)^7=-\omega^2=1+\omega. \] Therefore, when written as \( A+B\alpha+C\alpha^2 \) with \( \alpha=\omega \), \[ A=1,\quad B=1,\quad C=0. \] So \[ 5(3A-2B-C)=5\bigl(3\cdot1-2\cdot1-0\bigr)=\boxed{5}. \]

Answer 2

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 \( 5(3A - 2B - C) = 5 \).