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What is the remainder when \(x^{9999} + x^{8888} + x^{7777 }+\cdots+ x^{1111} + 1\) is divided by \(x^2 - 1\)?

 Jul 7, 2020
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Let Q(x) be the quotient and R(x) be the remainder:

 

\(x^{9999} + x^{8888} + x^{7777} + \cdots + x^{1111} + 1 = Q(x) (x^2 - 1) + R(x)\)

 

If we let x2 = 1 on both sides, then it gives \(x^{9999} + x^{8888} + x^{7777} + \cdots + x^{1111} + 1 = R(x)\) when all x2's are replaced by 1.

 

We rewrite the expression more explicitly in terms of x2.

 

\(x^{9999} + x^{8888} + x^{7777} + \cdots + x^{1111} + 1 = (x^2)^{4999}\cdot x+ (x^2)^{4444} + (x^2)^{3888}\cdot x + \cdots + (x^2)^{555}\cdot x + 1\)

 

Replacing all x2 by 1:

 

\(R(x) = x + 1 + x + 1 + x + 1 + x + 1 + x + 1 = \boxed{5x + 5}\)

 Jul 7, 2020

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