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Step-by-step Solution
Step 1: Recognize the equation satisfied by α
We know that α satisfies the equation
x^2 + x + 1 = 0.
Hence,
α^2 + α + 1 = 0.
Step 2: Express (1 + α)⁷ in a simpler form
We need to compute (1 + α)^7. Observe that
1 + α = -α^2
because
α^2 + α + 1 = 0 \implies 1 + α = -α^2.
Therefore,
(1 + α)^7 = (-α^2)^7 = - (α^2)^7 = - α^{14}.
Step 3: Simplify α^{14} using α^3 = 1
Since α^2 + α + 1 = 0 and α is a (non-real) cube root of unity, we have
α^3 = 1.
Thus,
α^{14} = α^{12} \cdot α^2 = (α^3)^4 \cdot α^2 = 1^4 \cdot α^2 = α^2.
Consequently,
(1 + α)^7 = - α^2.
Step 4: Rewrite - α^2 in terms of A + Bα + Cα^2
From α^2 + α + 1 = 0, we also get
α^2 = - (1 + α).
Hence,
- α^2 = 1 + α.
Therefore,
(1 + α)^7 = 1 + α,
i.e.,
A = 1,\; B = 1,\; C = 0.
Step 5: Calculate the required expression
We want 5(3A - 2B - C). Substituting A = 1,\; B = 1,\; C = 0, we get:
3A - 2B - C = 3(1) - 2(1) - 0 = 3 - 2 = 1.
Hence,
5(3A - 2B - C) = 5 \times 1 = 5.
Final Answer
The value of 5(3A - 2B - C) is 5.
