I want to factor the following polynomial: $$-x^3+x^2+5x+3$$
I started by using the rational root theorem and got the factors $\pm1$ and $\pm3$ for the constant. Via synthetic division i found out that $(x-3)$ is a factor. Now i'm stuck with the equation $(x-3)\cdot(-x^2-2x-1)$.
I tried using the sum of the factors of $-1$ but it turns out that they are $0$ and not $-2$.
$$-1=1\cdot(-1) \implies 1+(-1)=0$$
I read that if the sum is equal to the number in front of the $x^1$ (in my case $-2$) these numbers are the missing factors so for example $(x-1)\cdot(x+1)$.
I know that there are other ways to factor my polynomial, but I want to understand why it is not working in my case or what I am doing wrong.
Thanks for any type of help :)
The sum of the roots of a monic polynomial $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is $-a_{n-1}$, if the roots are counted with their multiplicities. In your case, you have $-x^2-2x-1$, which is not monic. But its roots are the roots of $x^2+2x+1$, which is monic. And this polynomial only has one root, $-1$, which is a double root. So, the sum of the roots, counted with their multiplicities, is $-2$.