Given the polynomial: $P(x) = x^7 + 9x^6 + 31x^5 + 55x^4 + 63x^3 + 55x^2 + 33x + 9 $ How do find it's roots ?
After a some time with try- error, I could verify: $ P(x) = (x + 1)^3 (x + 3)^2(x + i)(x − i) $
However is there a way to utilize the fact that all coefficient are positive, integer and that it is a monic polynominal?
As always any constructive help/hint/answer/recommendation for further reading is appreciated.
It's usually worth trying the rational root theorem which says that any rational root of the polynomial $$a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$$ can be written in the form $p/q$ where $p$ is a factor of $a_0$ and $q$ is a factor or $a_n$.
In your case this leads us to check for $\pm1$, $\pm3$, $\pm9$ as roots, and as Travis points out in the comments, the positive ones can be easily discarded. Presumably you would start with checking $-1$ and upon finding that it works you would factor out the $(x+1)$ term(s) so you don't have huge powers for subsequent checks.
After dealing with the rational roots in this case you get down to a quadratic - easy enough to solve the rest.
The key point here is to try finding rational roots first and then worry about other solutions.