# 2012 AMC 12B Problems/Problem 23

## Problem 23

Consider all polynomials of a complex variable, , where and are integers, , and the polynomial has a zero with What is the sum of all values over all the polynomials with these properties?

## Solution

First, assume that , so or . does not work because . Assume that . Then , we have , so a=44+b=a+c \leq a+b4+b=4+cb=c(a,b,c,d)(4,t,t,0)P(-1)=4-4+t-t+0=0P(1)\sum_{t=0}^{4} (4+4+t+t) = 40+20=60$.

Second, assume that$ (Error compiling LaTeX. ! Missing $ inserted.)z_0\in \mathbb{C} \backslash \mathbb{R}$.

==Solution (doubtful) ==

Since$ (Error compiling LaTeX. ! Missing $ inserted.)z_0PPz_0z_0z_0Pz_0\{\pm1,\pm i,(-1\pm i\sqrt{3})/2\}Pz+1z-1z^2+1z^2+z+1z=1a+b+c+d+4=0a,b,c,dz=-1(a+c)-(b+d)=4a+b=8,7,6,5,4b+d=4,3,2,1,0a,b,c,d4,4,4,04,3,3,04,2,2,04,1,1,04,0,0,0$.

Now suppose$ (Error compiling LaTeX. ! Missing $ inserted.)z=i4=(a-c)i+(b-d)a=cb-d=4a=b=cd=a-4a,b,c,d4,4,4,0$, which we have already counted in a previous case.

Suppose$ (Error compiling LaTeX. ! Missing $ inserted.)z=-i4=i(c-a)+(b-d)a=cb=4+da,b,c,d4,4,4,0$which we have previously counted.

Finally suppose$ (Error compiling LaTeX. ! Missing $ inserted.)z^2+z+1Pz^3=1b=4+ca,b,c,d4,4,0,0$.

Hence we've the polynomials <cmath>4x^4+4x^3+4x^2+4x</cmath> <cmath>4x^4+4x^3+3x^2+3x</cmath> <cmath>4x^4+4x^3+2x^2+2x</cmath> <cmath>4x^4+4x^3+x^2+x</cmath> <cmath>4x^4+4x^3</cmath> <cmath>4x^4+4x^3+4x^2</cmath> However, by inspection$ (Error compiling LaTeX. ! Missing $ inserted.)4x^4+4x^3+4x^2+4x+4x^4+x^3+x^2+x+1=(x^5-1)/(x-1)z_0z_0nn>5z_0nnn>5nppnz_0z_0P(z_0)=0(x^5-1)/(x-1)(x^5-1)/(x-1)=x^4+x^3+x^2+x+1PP(x)=4(x^4+x^3+x^2+x+1)$. This completes the proof.