Question

Find the remainder when x^100 - x^99 + x^98 − · · · − x + 1 is divided by x + 1.

Answer 1

The Answer is 101

Explanation:

101 is how much is remaining

Answer 2

The remainder is 101.

Explanation:

We are dividing:

`(x^(100)-x^(99)+x^(98)...-x+1)/(x+1)`

By the Polynomial Remainder Theorem, if we are dividing a polynomial P(x) by a binomial in the form (x - a), the our remainder will be P(a).

The divisor is (x + 1). Therefore, our a = -1.

Then the remainder will be:

`P(-1)`

We can rewrite our polynomial as:

`P(x)=-(x^(99)+x^(97)...+x^3+x)+(x^(100)+x^(98)...+x^2)+1`

Each of the parentheses contain fifty terms. -1 to any odd power is -1, and -1 to any even power is 1. Therefore:

`P(-1)=-(-50)+(50)+1`

Evaluate:

`P(-1)=101`

The remainder is 101.