Question

functions

Answer 1

\(\large \color{black}{\begin{align} & f(x)=x^4+x^3+x^2+x+1,\ \ x\in \mathbb{Z}^{>1} \hspace{.33em}\\~\\
& \normalsize \text{Find remainder when }\ \large f(x^5)\ \normalsize \text{is divided by} \ \large f(x) ? \hspace{.33em}\\~\\
& a.)\ 1 \hspace{.33em}\\~\\
& b.)\ 4 \hspace{.33em}\\~\\
& c.)\ 5 \hspace{.33em}\\~\\
& d.)\ \normalsize \text{a monomial in }\ x \hspace{.33em}\\~\\
& e.)\ \normalsize \text{a polynomial in }\ x \hspace{.33em}\\~\\
\end{align}}\)

Answer 2

Rewrite \(f(x) = \dfrac{x^5-1}{x-1}\) and \(f(x^5) = \dfrac{x^{25}-1}{x^5 - 1}\)

Answer 3

Don't we need \(|x|<1\) for that to be true? @ParthKohli

Answer 4

Nope, not at all.

Answer 5

Oh right I'm thinking of infinite sums.

Answer 6

http://www.wolframalpha.com/input/?i=table%5B%28x%5E%2825%29-1%29%2F%28x%5E5-1%29+mod%28%5Cfrac%7Bx%5E5-1%7D%7Bx-1%7D%29%2C%7Bx%2C2%2C10%7D%5D%3D

Answer 7

Notice that \(f(x) = \dfrac{x^5-1}{x-1} \implies x^5-1=f(x)*(x-1)\tag{1}\)


\[\begin{align}f(x)&=x^4+x^3+x^2+x+1 \\~\\
\implies f(x^5)&=x^{20}+x^{15}+x^{10}+x^5+1\\~\\
&=(x^{20}-1)+(x^{15}-1)+(x^{10}-1)+(x^5-1)+5\\~\\
&=(x^5-1)(stuff)+5\\~\\
&=f(x)*(x-1)*(stuff)+5 ~~~\color{gray}{\text{(from (1))}}\\~\\
&\equiv 0+5\pmod{f(x)}
\end{align}\]

Answer 8

wow thnx!

Answer 9

look up problem #30

Answer 10

oh u remember all things fro past cool.

Answer 11

*from

Answer 12

looks very handy book

Answer 13

i will solve it within 10 years. haha

Answer 14

try these
https://drive.google.com/folderview?id=0B8qeUE5SqcPAWFVaM1N5anN3S2M&usp=sharing

Answer 15

its showing "error 500"

Answer 16

try this
https://drive.google.com/drive/folders/0B8qeUE5SqcPAWFVaM1N5anN3S2M

Answer 17

it has a very good collection of olympiad textbooks

Answer 18

yes got it .