First if we know the roots W,X,Y,Z we can write the polynomial as
x^4 (W+X+Y+Z)x^3+(WX+WY+WZ+XY+XZ+YZ)x^2(WXY+WXZ+WYZ+XYZ)x+(WXYZ)
Now suppose:
W=a+bi
X=abi
Y=c+di
Z=cdi
It turns out that it doesn't matter which pair (W&Y, W&Z, X&Y, or X&Z) sums to 3+4i and its complement produces 13+i so let
W+Y=3+4i
XZ=13+i
Substituting the real and imaginary parts yields the system:
a+c=3
b+d=4
acbd=13
adbc=1
Solving (many steps omitted) yields a and c as the real solutions to
4a^4+24a^3113a^2+231a 104=0
Wolfram alpha gives these as (6±
√(2√(4265)59))/4
(more omitted)
The approximate roots of F are
W=.6121.942i
X=.612+1.942i
Y=2.388+5.942i
Z=2.3885.942i
Returning to the very top we can recreate F.
F(x)=x^46x^3+
51x^270x+170

Posted by Jer
on 20160207 16:12:32 