Question

How many complex roots does the equation below have?

x 6 + x 3 + 1 = 0

The number of complex roots is

Answer 1

6

Explanation:

Answer 2

The number of complex roots is 6.

Explanation:

Descartes's rule of signs tells you that the number of positive real roots is 0. The number of negative real roots will be at most 2. The minimum value of the left side will be between x=0 and x=-1, but will never be negative. Thus all six roots are complex.

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The magnitude of x^3 will exceed the magnitude of x^6 only for values of x between -1 and 1. Since the magnitude of either of these terms will not be more than 1 in that range, the left-side expression must be positive everywhere.