By Descartes' Rule of Signs:
The signs from the original equation are: + - + - - +. 3 sign changes mean that there are either 3 or 1 positive roots.
If we change the signs of the odd-powered terms: + + + + - -. This 1 sign change means that there is exactly 1 negative root.
All roots:
We can immediately factor out x from the equation:
x (x^5 - 3x^4 + 2x^3 - 6x^2 - 15x + 45) = 0
Factor out (x-3) since x = 3 is a root:
x (x - 3) (x^4 + 2x^2 - 15) = 0
Factor the last term further:
x (x - 3) (x^2 - 3) (x^2 + 5) = 0
This allows us to determine the rest of the x-values:
x = 0, 3, sqrt3, -sqrt3, i*sqrt5, i*-sqrt5
