BdMO National Higher Secondary 2007/7
BdMO National Higher Secondary 2007/7
Problem 7:
$f(x)=x^6+x^5+\cdots +x+1$Find the remainder when dividing $f(x^7)$ by $f(x)$.
$f(x)=x^6+x^5+\cdots +x+1$Find the remainder when dividing $f(x^7)$ by $f(x)$.

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Re: BdMO National Higher Secondary 2007/7
I am quite sure that the solution is the following:
f(x)= 1+x+x^2+x^3+.....+x^6
Now if we write it like this:
f(x)= 1+x+x^2+x^3+.....infinite, then
f(x)= (1x)^(1)
f(x^7)=(1x^7)^(1)
so f(x^7)/f(x) = (1x)/(1x^7)
according to remainder theorem, the remainder is = 0
so Answer: 0.
(Osman, a friend of mine solved this)
f(x)= 1+x+x^2+x^3+.....+x^6
Now if we write it like this:
f(x)= 1+x+x^2+x^3+.....infinite, then
f(x)= (1x)^(1)
f(x^7)=(1x^7)^(1)
so f(x^7)/f(x) = (1x)/(1x^7)
according to remainder theorem, the remainder is = 0
so Answer: 0.
(Osman, a friend of mine solved this)
Re: BdMO National Higher Secondary 2007/7
Actually that is applicable when x is less than 1 but this was not given in the question, and I'm looking for a solution of this one long time so I think some of the seniors could help
 nafistiham
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Re: BdMO National Higher Secondary 2007/7
Hint:
solution
\[\sum_{k=0}^{n1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.
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Re: BdMO National Higher Secondary 2007/7
nice and clean:D

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Re: BdMO National Higher Secondary 2007/7
How does the green line lead to the red line?nafistiham wrote: here, $x^71=f(x)\cdot(x1)$
so, \[f(x)(x^{42}1),(x^{35}1),(x^{28}1),(x^{21}1),(x^{14}1),(x^71)\]
so, the remainder will be
\[7\]
 nafistiham
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Re: BdMO National Higher Secondary 2007/7
actually, what I didn't think much important to mention was thissakib.creza wrote:How does the green line lead to the red line?nafistiham wrote: here, $x^71=f(x)\cdot(x1)$
so, \[f(x)(x^{42}1),(x^{35}1),(x^{28}1),(x^{21}1),(x^{14}1),(x^71)\]
so, the remainder will be
\[7\]
\[f(x)x^{7}1x^{7n}1\]
It should be clear enough now
\[\sum_{k=0}^{n1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.