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.

Let r=(i*sqrt(3)-1)/2 and s=(-i*sqrt(3)-1)/2

P(x)=Q(x)(x-r)(x-s) + ax+b where Q is the quotient and ax+b is the remainder.

P(r)=ar+b  and  P(s)=as+b -->

a = ( P(r)-P(s) )/(i*sqrt(3))

b = ( P(r)+P(s)+a )/2

For example, with P(x)=2x^5 - 3x^2 + x + 4

P(r)=4+sqrt(3)*i  and  P(s)=4-sqrt(3)*i -->

a=2  and  b=5 --> remainder of 2x+5

Of course, you're going to want to use at least a graphing calculator for the computations.

Posted by Dennis on 2008-04-17 11:12:52