17. Suppose that the polynomial $P(x)=x^{3}+4 x^{2}+b x+c$ has a single root $r$ and a double root $s$ for some distinct real numbers $r$ and $s$. Given that $P(-2 s)=324$, what is the sum of all possible values of $|c|$ ?
17. 108
By Vieta's formula, we have $r+2 s=-4$ and writing $P(x)=(x-r)(x-s)^{2}$, we have $324=P(-2 s)=(-2 s-r)(-3 s)^{2}=36 s^{2}$. Thus, $s^{2}=9$ and $s \in\{-3,3\}$. We next observe that $|c|=|P(0)|=|r| s^{2}=|-4-2 s| s^{2}$. Hence, the sum of all possible values of $|c|$ is $9(|-4-6|+|-4+6|=108$