Question

How many solutions over the complex number system does this polynomial have?

Answer 1

using Descartes Rule of signs \(\bf \textit{Descartes Rule of Signs}\\
{\color{blue}{ +}}2x^4{\color{blue}{ -}}3x^3{\color{blue}{ -}}24x^2{\color{blue}{ +}}13x{\color{blue}{ +}}12=f{\color{brown}{ (x)}}
\\ \quad \\ \quad \\
{\color{blue}{ +}}2x^4{\color{blue}{ +}}3x^3{\color{blue}{ -}}24x^2{\color{blue}{ -}}13x{\color{blue}{ +}}12=f{\color{brown}{ (-x)}}\)

notice the sign changes

twice for f(x), thus real positive are 2,0
twice for f(-x), thus real negative are also 2,0

4th degree polynomial, means 4 roots

either 2 positive and 2 negative
or
2 positive, 0 negative and thus 2 complex
or
0 positive, 2 negative and thus room for 2 complex again

Answer 2

WOw thank you so much!

Answer 3

yw