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=4
4+b=a+c \leq a+b
4+b=4+c
b=c
(a,b,c,d)
(4,t,t,0)
P(-1)=4-4+t-t+0=0
P(1)
\sum_{t=0}^{4} (4+4+t+t) = 40+20=60$.
Second, assume that$ (Error compiling LaTeX. Unknown error_msg)z_0\in \mathbb{C} \backslash \mathbb{R}$.
==Solution (doubtful) ==
Since$ (Error compiling LaTeX. Unknown error_msg)z_0P
P
z_0
z_0
z_0
P
z_0
\{\pm1,\pm i,(-1\pm i\sqrt{3})/2\}
P
z+1
z-1
z^2+1
z^2+z+1
z=1
a+b+c+d+4=0
a,b,c,d
z=-1
(a+c)-(b+d)=4
a+b=8,7,6,5,4
b+d=4,3,2,1,0
a,b,c,d
4,4,4,0
4,3,3,0
4,2,2,0
4,1,1,0
4,0,0,0$.
Now suppose$ (Error compiling LaTeX. Unknown error_msg)z=i4=(a-c)i+(b-d)
a=c
b-d=4
a=b=c
d=a-4
a,b,c,d
4,4,4,0$, which we have already counted in a previous case.
Suppose$ (Error compiling LaTeX. Unknown error_msg)z=-i4=i(c-a)+(b-d)
a=c
b=4+d
a,b,c,d
4,4,4,0$which we have previously counted.
Finally suppose$ (Error compiling LaTeX. Unknown error_msg)z^2+z+1P
z^3=1
b=4+c
a,b,c,d
4,4,0,0$.
Hence we've the polynomials
<cmath>4x^4+4x^3+4x^2+4x</cmath>
<cmath>4x^4+4x^3+3x^2+3x</cmath>
<cmath>4x^4+4x^3+2x^2+2x</cmath>
<cmath>4x^4+4x^3+x^2+x</cmath>
<cmath>4x^4+4x^3</cmath>
<cmath>4x^4+4x^3+4x^2</cmath>
However, by inspection$ (Error compiling LaTeX. Unknown error_msg)4x^4+4x^3+4x^2+4x+4x^4+x^3+x^2+x+1=(x^5-1)/(x-1)
z_0
z_0
n
n>5
z_0
n
n
n>5
n
p
p
n
z_0
z_0
P(z_0)=0
(x^5-1)/(x-1)
(x^5-1)/(x-1)=x^4+x^3+x^2+x+1
P
P(x)=4(x^4+x^3+x^2+x+1)$. This completes the proof.