Math  /  Algebra

Question13. Solve z6+z3+1=0z^{6}+z^{3}+1=0 (hint: solve z91=0z^{9}-1=0 first).

Studdy Solution
Verify the solutions:
The solutions are z=e2πik/9 z = e^{2\pi i k / 9} for k=1,2,4,5,7,8 k = 1, 2, 4, 5, 7, 8 .
These correspond to:
z=e2πi/9,e4πi/9,e8πi/9,e10πi/9,e14πi/9,e16πi/9 z = e^{2\pi i / 9}, e^{4\pi i / 9}, e^{8\pi i / 9}, e^{10\pi i / 9}, e^{14\pi i / 9}, e^{16\pi i / 9}
These are the roots of the equation z6+z3+1=0 z^6 + z^3 + 1 = 0 .
The solutions are:
z=e2πi/9,e4πi/9,e8πi/9,e10πi/9,e14πi/9,e16πi/9 z = e^{2\pi i / 9}, e^{4\pi i / 9}, e^{8\pi i / 9}, e^{10\pi i / 9}, e^{14\pi i / 9}, e^{16\pi i / 9}

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