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This essential trains for: AMC-10, AMC-12, AIME.
The cubic roots of unity are the solutions of the polynomial equation:
This is a cyclotomic equation - see the lesson on cyclotomic equations.
We expect it to have three complex roots, since its degree is 3.
To solve this equation, write the number 1 in trigonometric form:
Take the third root by applying the de Moivre formula:
Since the expression of the roots is periodic, there are three distinct values for the roots, given by k=0, 1, 2:
However, it is easy to see that the third root is the complex conjugate of the second one, due to elementary trigonometric transformations:
Traditionally, the complex-imaginary roots of unity have been denoted with ω :
Therefore, the roots of one are:
Since, according to De Moivre's formula:
we can also write them as:
Which is pretty obvious when we think of the algebraic identity:
which shows by factoring that the one real root is equal to 1 and the two complex-imaginary roots are roots of the quadratic equation:
Properties of the complex roots of unity
From the above discussion:
Problem (similar to a problem from a 1959 Moscow Olympiad), illustrating the connection to polynomials:
If the polynomial
is expanded into a canonical form:
what is the value of the expression:
Solution We notice that:
and so on. The real parts of these powers of ω form the sequence:
and the imaginary parts form the sequence:
The sequence of real parts corresponds exactly to the coefficients of the expression we have to compute. We realize that the expression can be computed by calculating the value of the polynomial at ω and taking the real part:
To compute, we use the properties of the cubic roots of unity:
and the that could be written for the imaginary part is zero for this particular polynomial: