<p>If \(\omega \neq 1\) is a cube root of unity, and \((1 + \omega)^7 = A + B\omega\). Then \((A, B)\) equals</p>
Step-by-Step Solution
Key Concept: Use the fundamental property that 1 + ω + ω² = 0 (so 1 + ω = -ω²) and apply the binomial expansion or recognize the pattern in powers of (1 + ω) using this substitution.
<p><strong>Step 1:</strong> Recall that ω is a cube root of unity with ω ≠ 1, so ω³ = 1 and 1 + ω + ω² = 0.</p><p><strong>Step 2:</strong> From 1 + ω + ω² = 0, we get 1 + ω = -ω².</p><p><strong>Step 3:</strong> Therefore, (1 + ω)⁷ = (-ω²)⁷ = -ω¹⁴.</p><p><strong>Step 4:</strong> Since ω³ = 1, reduce the exponent: ω¹⁴ = ω^(3·4 + 2) = (ω³)⁴ · ω² = 1⁴ · ω² = ω².</p><p><strong>Step 5:</strong> Thus (1 + ω)⁷ = -ω².</p><p><strong>Step 6:</strong> From 1 + ω + ω² = 0, we have ω² = -1 - ω, so -ω² = 1 + ω.</p><p><strong>Step 7:</strong> Alternatively, express -ω² in the form A + Bω. Since ω² = -1 - ω, we get -ω² = 1 + ω.</p><p><strong>Step 8:</strong> Therefore (1 + ω)⁷ = 1 + ω, giving A = 1 and B = 1.</p><p>∴ Answer: (A, B) = (1, 1)</p>
Correct Answer: C