In this question A^B means A is raise to the power B.let f(x)=1+x+x^2+x^3+.........+x^6.The reminder when f(x^7) is divided by f(x) is?
(a). 0
(b).6
(c).7
(d).none

• reminder will be 7
• 8 years agoHelpfull: Yes(9) No(1)
• (1+x^7 + x^14 + x^21 .........+ x^42)/ (1+x+x^2+x^3.......x^6)
  
  =(sum of gp with first term 1 and common ratio x^7)/(sum of go with first term 1 and common ratio x)
  
  =[1*(x^49 - 1)/(x-1)] / [1*(x^7 - 1)/(x-1)]
  =(x^49 - 1)/(x^7 - 1)
  
  According to remainder theorem,
  Deno=0
  x^7 - 1=0 , x=1
  
  Put in numerator,
  Remainder= 1^49 -1=0
• 8 years agoHelpfull: Yes(6) No(12)
• If we substitute x with 0 or 1 then the remainder will be 0
  
  put x = 0,1 or any value
  
  put x=0
  f(x)=1
  f(x^7)=1
  f(x^7)/f(x)= 1/1 => remainder = 0
  
  put x=1
  f(x)=7
  f(x^7)=7
  f(x^7)/f(x)= 7/7 => remainder = 0
  
  
  But if the value of x>1
  then it will produce 7 as the remainder
  
  There is a pattern.
  if , f(x) = 1+x+x^2+.....+x^(n).
  for x > 1,
  then the remainder when f(x^(n+1)) is divided by f(x) is (n+1).
  so , here remainder is 7.
  
  
  I think they must have given the conditions where x>1 or something else to predict the correct result...
  
• 8 years agoHelpfull: Yes(3) No(0)
• Multiply the f(x)=1+x+x^2+...x^6 equation with x^6 then we will get the equation of function f(x^7)=x^6+x^7+......+x^12. On dividing we get remainder 0.
• 8 years agoHelpfull: Yes(1) No(2)
• answer will be a
• 8 years agoHelpfull: Yes(0) No(4)
• right answer is a
• 8 years agoHelpfull: Yes(0) No(2)
• sum of gp with first n terms with common ratio x^7 is 1*(x^49 - 1)/(x^7-1) but not 1*(x^49 - 1)/(x-1)
• 8 years agoHelpfull: Yes(0) No(0)
• f(x7)=1+x7+(x7)2 + ....+ (x7)6 = 1+x7+x14+....+x42
  We will rewrite the above equation, f(x7)=1+(x7−1)+(x14−1)+... + (x42−1)+6
  We know that x7−1=(x−1)(x6+x5+...1)
  (∵ xn−an = (x−a).(xn−1+xn−2.a+xn−3.a2.....+an−1 )
  Now It is clear that x7−1 is exactly divisible by f(x).
  Also x14−1=(x7)2−12 and x7−1 is a factor of this expression. (∵xn−an is always divisible by x−a
  Similarly, we write x21−1=(x7)3−13, x28−1=(x7)4−14....
  So remainder = 1 + 6 = 7
  
• 8 years agoHelpfull: Yes(0) No(1)
• f(x^7) = 1+x^7( f(x) )
  
  1+x^7( f(x) ) / f(x)
  
  is of the form (ax+1)/a
  So ans is 1
• 8 years agoHelpfull: Yes(0) No(2)
• (c) .7 is correct ans.
• 8 years agoHelpfull: Yes(0) No(0)