Question

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

Answer 1

Answer:

The remainder is 101.

Step-by-step explanation:

We are dividing:

$(x^{100}-x^{99}+x^{98}...-x+1)\div(x+1)$

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.

Answer 2

Answer:

The Answer is 101

Step-by-step explanation:

101 is how much is remaining