I am trying to find the roots of characteristic polynomials.
This can be difficult to do by hand, especially when I have a characteristic polynomial of the 3rd or 4th degree.
I always try to factorise the polynomial, but it doesn't always work.
I start by guessing simple roots like 1,2,-1,-2...
I want help in finding the roots in a shorter period of time.
You have the rational root test.
If you have a polynomial of this type:
${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0}$ with integer coefficients ${\displaystyle a_{i}\in \mathbb {Z} }$.
The test tells you that each rational solution $x = \frac{p}{q}$, written in lowest terms so that p and q are relatively prime, satisfies:
p is an integer factor of the constant term $a_0$, and q is an integer factor of the leading coefficient $a_n$.
All the rational (racionals and integers) roots are of this form.
In other words, if there is a rational root in can be written like $x = \frac{p}{q}$ such that p divides $a_0$ and q divides $a_n$.
If you try all of these combinations of numerators and denominators and none of them is a root, then there are no roots in $\mathbb{Q}$.