Problem:
Let P be the product of the nonreal roots of x4−4x3+6x2−4x=2005. Find ⌊P⌋. (The notation ⌊P⌋ denotes the greatest integer that is less than or equal to P.)
Solution:
The given equation is equivalent to x4−4x3+6x2−4x+1=2006, that is, (x−1)4=2006. Thus (x−1)2=±2006, and x−1=±42006 or ±i42006. Therefore the four solutions to the given equation are 1±42006 and 1±i42006. Then P=(1+i42006)(1−i42006)=1+2006, so ⌊P⌋=45.