There's probably a more clever method than just multiply out, set to zero, and refactor. But it worked. I'd like to see a better way, but this was a nice diversion.
The product is
864z^4+720z^3+210z^2+25z+1=2
set to zero
864z^4+720z^3+210z^2+25z0=0
I figured if this is solvable its probably the product of two quadratics so I decided to try factoring by hand as
(az^2+bz+1)(dz^2+ez1)=864z^4+720z^3+210z^2+25z1
which leads to the system
ad=864
ae+bd=720
a+be+d=210
b+e=25
solving for b gives
b=(720a25a^2)/(a^2+864)
which only has a few solutions with positive integer for a and only one of these gives positive integers for the rest. The factorization sought is:
(12z^2+5z+1)(72z^2+30z1)=0
and the quadratic formula finishes us off. The solutions are:
(5±i√23)/24 and (5±√33)/24

Posted by Jer
on 20160403 20:19:34 