2010 AIME II Problems/Problem 7

Problem 7

Let $P(z)=x^3+ax^2+bx+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$.

Solution

set $w=x+yi$, so $x_1 = x+(y+3)i$, $x_2 = x+(y+9)i$, $x_3 = 2x-4+2yi$.

Since $a,b,c\in{R}$, the imaginary part of a,b,c must be 0.

Start with a, since it's the easiest one to do: $y+3+y+9+2y=0, y=-3$

and therefore: $x_1 = x$, $x_2 = x+6i$, $x_3 = 2x-4-6i$

now, do the part where the imaginery part of c is 0, since it's the second easiest one to do: $x(x+6i)(2x-4-6i)$, the imaginery part is: $6x^2-24x$, which is 0, and therefore x=4, since x=0 don't work

so now, $x_1 = 4, x_2 = 4+6i, x_3 = 4-6i$

and therefore: $a=-12, b=84, c=-208$, and finally, we have $|a+b+c|=|-12+84-208|=136$.