Find the number of complex roots, possible number of real roots, imaginary roots and possible rational roots.

Question

Equation is 8 - 4x^3 + 4x^6 = 0

Answer 1

Let f(x) = 4x⁶-4x³+8

 

f(x) has 2 sign changes.  So, by Descartes Rule of Signs, f(x) has either                                                    2 positive roots or none.

 

f(-x) = 4(-x)⁶-4(-x)³+8 = 4x⁶+4x³+8

 

f(-x) has no sign changes.  So, by Descartes Rule of Signs, f(x) has no                                                    negative roots.

 

f(x) has degree 6.  So, counting multiplicities, there are a total of 6                                                                                               roots.

 

First possibility:  2 positive roots   0 negative roots  4 imaginary roots

 

Second possibility:  0 positive   0 negative   6 imaginary