Question

What would be the Remainder when
1+x+x^2+x^3+.... x^2021 is divided by x+1.

Answer 1

when 1+x+x^2+x^3+...x^2021 is divided by (x+1) the remainder will be p(-1)

so p(-1)= 1+(-1)+(-1)^2+(-1)^3+...(-1)^2021

= 0 [ as (-1)^n is -1 when n is odd and (-1)^n is 1 when n is even.

so you can see in above expression that,

1-1+1-1+1-1.... continues upto (-1)^2021 and positive and negative 1 cancel each other so at the last remainder becomes 0

hope it helps you

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