Let z1, z2, z3 and z4 be the roots of the equation z7 + z4 + z3 + z + 1 = 0 such that
zi3 + 1 zi2 i {1, 2, 3, 4}. If area of the quadrilateral formed by z1, z2, z3 and z4 on argand plane is (p cos3q°) where q (0, 90) then find the value of
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Given the equation z⁷ + z⁴ + z³ + z + 1 = 0. Note that z⁵ - 1 = (z-1)(z⁴+z³+z²+z+1). Multiplying both sides by (z²-1) gives (z⁷ - z²) - (z⁵ - 1) = (z²-1)(z⁵-1). Rearranged, this is z⁷ - z⁵ - z² + 1 = (z²-1)(z⁵-1). Our equation is z⁷ + z⁴ + z³ + z + 1 = 0. Adding these two equations yields 2z⁴ + 2z³ + 2z + 2 = 2(z⁴+z³+z+1)=0, so z⁴+z³+z+1=0. Factoring, (z⁴+1) + z(z²+1)=0. The roots are the 4 primitive 8th roots of unity, excluding ±i: e^(±iπ/4), e^(±i3π/4). These form a square on the Argand plane with vertices at 45°, 135°, 225°, 315°. The side length is √2, so area is (√2)² = 2. Thus, p cos³q° = 2. Since cos60°=1/2, cos³60°=1/8, so p=16 and q=60. Therefore, (p+q)/5 = (16+60)/5 = 76/5 = 15.2.
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