Devise an algorithm which, for any polynomial P(x), will determine the polynomial remainder of P(x)/(x^2+x+1) without actually performing the division.

(x + 1)²  = x² +2x + 1
  Coeff's:  (B)1  (C)2  (A)1
  Rem = (C - B) * x + (A - B)
         = (2 - 1) * x + (1 - 1)
         = x
(x + 1)²/(x² +x + 1) =  1 + x/(x² +x + 1) ;
                               [1 * x  being the remainder]

(x + 1)³ = x³ + 3x² + 3x + 1
  Coeff's:  (B)3 (C) 3  (A) 1 + 1 [ie, first and last]
  Rem = (C - B) * x + (A -B)
         = (3 - 3) * x + (2 -3)
         =  -1
(x + 1)³/(x² +x + 1) = x + 2  -1/(x² +x + 1)

Posted by brianjn on 2008-04-16 21:10:09