Problem: Sum of absolute values of polynomial roots Can you please give me some hints as to how I might approach this problem? Thanks!

Given the polynomial $f = 2X^3 - aX^2 - aX + 2, a \mathbb \in R$ and roots $x_1, x_2$ and $x_3,$ find $a$ such that $|x_1| + |x_2| + |x_3| = 3.$

Edit: We know $-1$ is one of the roots of that polynomial, regardless of the value of $a$. So, in essence, what we have to demonstrate is that $|x_2| + |x_3| = 2.$

We first have the following factorization,
$ 2x^3 - ax^2 - ax + 2 = (x+1)(2x^2 - (2+a)x + 2).$
Suppose $x_1 = -1$, then $|x_2| + |x_3| = 2$ and they are solutions to the quadratic equation $2x^2 - (2+a)x + 2 = 0$. Hence,
$ x_2 + x_3 = 1 + a/2 $ and 
$ x_2 x_3 = 1$.
By observing the second equation, $x_2$ and $x_3$ are either both positive or both negative, so we have
$1 + a/2 = x_2 + x_3 = 2$ or $-2$. 
Therefore, $a = 2$ or $-6$.
