Question

Find the remainder when 1+x+x^2+....+x^2017 is divided by (x-1)

Answer 1

Step-by-Firstly, I want to say that when divisor has degree n, remainder must have a degree of less than n. This concept is essential to solve the problem.

Now, let f(x) = x^2017 = (x^2 - 3x + 2 ) g(x) + (ax + b)

Here, g(x) is quotient and (ax + b) being the required remainder of degree less than 2.

It may be linear polynomial or even be a constant of degree 0 if a will be equal to 0. So, its degree is either 1 or 0 and we have to find a and b to predict the remainder.

Now, f(x) = x^2017 = (x-2) (x-1)g(x) + (ax + b) by factorizing

Now, f(1) = 1^2017 = (1-2) (1-1)g(x) + (a.1 + b) = a+b

or, a + b = 1 …….. eqn 1

similarly, f(2) = 2^2017 = (2-2) (2-1)g(x) + (a.2 + b) = 2a+b

or, 2a + b = 2^2017………….. eqn 2

eqn 2 - eqn 1 gives a = 2^2017 - 1

by putting it in eqn 1 = 2^2017 - 2

so the required remainder is (ax + b) = (2^2017 - 1)x + 2^2017 - 2tep explanation: