What is the least number of imaginary roots does this equation have $$x^{10} - 4x^6 + x^4- 2x -3 = 0$$
Answer is - at least four imaginary roots
I tried to use Descartes rule of signs, but since the powers are not continuously decreasing here, I cannot use it. Please tell me, if there is any other way to solve this problem.
If $x= ai$ where $a$ is real then we get $$ -a^{10}+4a^6+a^4-2ai-3=0$$
Since both real and imaginary parts are $0$ we get $a=0$ and so $-3=0$. Thus this equation has no imaginary solution.
Anyway, this equation has $10$ complex solutions!
Of course, the number of real solutions is most interesting to find.