P(x)= x^2012+x^2011+x^2010+...+x+1)^2 - x^2012 Q(x)=x^2011+x^2010+..+x+1 The remainder when P(x) is divided by Q(x) is ?

• Put x=1
  P(1)= 2013^2-1
  Q(1)=2012 = 2013-1
  P(1) divided by Q(1) is remainder 0.
• 11 years agoHelpfull: Yes(17) No(2)
• P(x) =[x^2012 + Q(x)]^2-x^2012
  
  P(x) =x^4024+Q^2+2*Q*x^2012 -x^2012
  P(x) =Q^2+2*Q*x^2012+(x^2012)*(x^2012-1)
  
  Now x^2012-1 = Q*(x-1)
  
  P(x) = Q^2 +2Qx^2012 +x^2012*Q*(x-1)
  
  As all terms of P are having Q as a factor,
  so P is fully divisible bt Q.
  
  so zero remainder.
• 11 years agoHelpfull: Yes(6) No(1)
• @ Karteek,
  
  What is the fun in copy pasting the puzzle and solution yourself?
  
  I do not think admin is giving marks in such cases.
  
  Enjoy and have fun.:-)
• 11 years agoHelpfull: Yes(4) No(1)
• p(x)=x^2012+x^2011+....+x+1-x^2012
  so x^2012 must be cut (x^2012-x^2012)=0
  p(x)=x^2011+....+x+1
  q(x)=x^2011+...+x+1
• 11 years agoHelpfull: Yes(3) No(0)
• P(x)= x^2012+x^2011+x^2010+...+x+1)^2 - x^2012
  Q(x)= x^2011+x^2010+..+x+1
  Q(x) = sum of 2012 terms of G.P
  = 1*(x^2012 - 1)/(x - 1)
  so
  P(x)= (x*Q(x)+1)^2 - x^2012
  then
  P(x)/Q(x) = (x*Q(x)+1)^2 - x^2012 / Q(x)
  
  (x*Q(x)+1)^2 can be written as:
  (x*Q(x))^2 + 2*x*Q(x) + 1
  
  P(x)/Q(x) = ((x*Q(x))^2 + 2*x*Q(x) + 1 - x^2012) / Q(x)
  so finally take a look :
  P(x)/Q(x) = (1 - x^2012) / Q(x)
  
  P(x)/Q(x) = (1 - x^2012) / (x^2012 - 1) / (x - 1)
  
  so clearly is getting P(x)/Q(x) = (1-x)
  enjoy.....
• 11 years agoHelpfull: Yes(1) No(9)
• Remainder will be zero.
  
  Let others try.
  
  I shall give details later.
• 11 years agoHelpfull: Yes(1) No(3)