What is the remainder when \(x^{9999} + x^{8888} + x^{7777 }+\cdots+ x^{1111} + 1\) is divided by \(x^2 - 1\)?
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:
\(R(x) = x + 1 + x + 1 + x + 1 + x + 1 + x + 1 = \boxed{5x + 5}\)