Question

Find the remainder when 1 + X + x square +...... + x power 2017 is divided by X - 1

Answer 1

x^2-1= (x+1)(x-1).

Now, let,

x^2007 = q(x) * (x+1)(x-1) * ax+b

where q(x)= quotient

And ax+b= Remainder in variable 'x'

Let x=1

1^2007 = a(1) + b

=> 1 = a+b------I

Also, x=-1

(-1)^2007 = a(-1) +b

=> 1 = a-b ------II

Solving I and II,

a=1 and b=0

Putting a=1 & b=0 in the remainder,

ax+b = x

So remainder =x