Find the remainder when x¹⁰⁰ is divided by x² − 1.

1. Write x¹⁰⁰ = (x² − 1)q(x) + ax + b. At x = 1: a + b = 1. At x = −1: −a + b = 1. So a = 0, b = 1 and the remainder is 1.

Find the remainder when $x^{100}$ is divided by $x^{2} - 1$ .

Competition practiceAlgebra

Write

x^{100} = (x^{2} - 1)

(x) +

ax +

b

. At

x = 1 : a + b = 1

. At

x = - 1 : - a + b = 1

. So

a = 0, b = 1

and the remainder is 1.

Answer: A) 1

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