Remainder when $P(x^5)$ is divided by $P(x) = x^4 + x^3 + x^2 + x + 1$ is _______.
Step-by-Step Solution
Key Concept: The remainder theorem combined with factoring expressions of the form $x^n - 1$ allows us to extract the constant remainder.
Given $P(x^5) = (x^{20}-1) + (x^{12}-1) + (x^{10}-1) + (x^2-1) + 5$, we determine that $P(x) = \frac{x^4-1}{x-1}$. When dividing $P(x^5)$ by $P(x)$, the remainder equals $5$ since all the difference terms are divisible by $P(x)$.
Correct Answer: 5