Question

Consider a function $f(x) = x^4 + x^3 + x^2 + x + 1$, where x is a positive integer greater than 1. What will be the remainder if $f(x^5)$ is divided by $f(x)$?

Accepted Answer

A. $f(x) = \frac{x^5 - 1}{x - 1} \implies x^5 \equiv 1 \pmod{f(x)}$. $f(x^5) = (x^5)^4 + (x^5)^3 + (x^5)^2 + (x^5) + 1$. Since $x^5 \equiv 1 \pmod{f(x)}$, $f(x^5) \equiv 1^4 + 1^3 + 1^2 + 1 + 1 = 5 \pmod{f(x)}$. So remainder is 5.

Polynomial Remainder Theorem

Explanation

Practise this under exam conditions