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Question
A root of unity is a complex number that is a solution to the equation, zn=1 for some positive integer n. Number of roots of unity that are also the roots of the equation z2+az+b=0, for some integer a and b is?
A root of unity is a complex number that is a solution to the equation, zn=1 for some positive integer n. Number of roots of unity that are also the roots of the equation z2+az+b=0, for some integer a and b is?
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Solution
The correct option is A 8
8.
The only real roots
of unity are 1 and −1. If ζ is a complex root of unity that is also a root of
the equation z2+az+b, then its conjugate ¯ζ must also be a root.
In this case, |a|=|ζ+¯ζ|≤|ζ|+|¯ζ|=2 and b=ζ¯ζ=1. So we only need to check
the quadratics z2+2z+1,z2+z+1,z2+1,z2−z+1,z2−2z+1.
We find 8 roots of unity:±1,±i,12(±1±√3i).
8.
The only real roots of unity are 1 and −1. If ζ is a complex root of unity that is also a root of the equation z2+az+b, then its conjugate ¯ζ must also be a root.
In this case, |a|=|ζ+¯ζ|≤|ζ|+|¯ζ|=2 and b=ζ¯ζ=1. So we only need to check the quadratics z2+2z+1,z2+z+1,z2+1,z2−z+1,z2−2z+1.
We find 8 roots of unity:±1,±i,12(±1±√3i).
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