JEEOlympiad
Let zeta_1, zeta_2, ..., zeta_8 be the eight primitive 15th roots of…
Question
Let zeta_1, zeta_2, ..., zeta_8 be the eight primitive 15th roots of unity (the complex numbers zeta with zeta^15 = 1 whose order is exactly 15). Compute the product (2 - zeta_1)(2 - zeta_2)...(2 - zeta_8). Report the exact integer value.
✓ Verified answer: 151checked by our engine — not a guess
Step-by-step solution
The primitive 15th roots of unity are exactly the roots of the 15th cyclotomic polynomial Phi_15(x), so prod (2 - zeta_j) = Phi_15(2).
Since 15 = 3*5, Phi_15(x) = (x^15 - 1)(x - 1)/[(x^5 - 1)(x^3 - 1)] which simplifies to the degree-8 polynomial x^8 - x^7 + x^5 - x^4 + x^3 - x + 1.
Evaluate at x = 2: 256 - 128 + 32 - 16 + 8 - 2 + 1 = 151.
(Check: 256 - 128 = 128; 128 + 32 = 160; 160 - 16 = 144; 144 + 8 = 152; 152 - 2 = 150; 150 + 1 = 151.)
Final answer151
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