Using the fifth roots of unity to find the roots

g2esebyy7

g2esebyy7

Answered question

2022-04-20

Using the fifth roots of unity to find the roots of (z+1)5=(z1)5

Answer & Explanation

Alexis Wolf

Alexis Wolf

Beginner2022-04-21Added 13 answers

Having calculated the fifth roots of unity 1,w,w2,w3,w4, i.e. the solutions to the equation u5=1, we can recast our equation as (z+1z1)5=1 provided z is not equal to 1
It is easy to verify that z=1 is not a solution to the equation (z+1)5=(z1)5. So for z not equal to 1, we have that z+1z1=1,w,w2,w3,w4. Simple manipulation will allow you to express z in terms of these roots.
Note that setting z+1z1=1 will not yield a solution. This can be expected, since the equation we are solving is quartic and so has at most 4 distinct roots.

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