Problem 7:
$f(x)=x^6+x^5+\cdots +x+1$Find the remainder when dividing $f(x^7)$ by $f(x)$.
BdMO National Higher Secondary 2007/7
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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)= (1-x)^(-1)
f(x^7)=(1-x^7)^(-1)
so f(x^7)/f(x) = (1-x)/(1-x^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)= (1-x)^(-1)
f(x^7)=(1-x^7)^(-1)
so f(x^7)/f(x) = (1-x)/(1-x^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
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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, $x^7-1=f(x)\cdot(x-1)$
so, \[f(x)|(x^{42}-1),(x^{35}-1),(x^{28}-1),(x^{21}-1),(x^{14}-1),(x^7-1)\]
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, $x^7-1=f(x)\cdot(x-1)$
so, \[f(x)|(x^{42}-1),(x^{35}-1),(x^{28}-1),(x^{21}-1),(x^{14}-1),(x^7-1)\]
so, the remainder will be
\[7\]
\[f(x)|x^{7}-1|x^{7n}-1\]
It should be clear enough now
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.