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#### melese

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- Feb 24, 2012

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**BGR,1989**) Prove that for any integer $n>1$ the equation $\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0$ has no rational roots.

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- Jan 25, 2013

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BGR,1989) Prove that for any integer $n>1$ the equation $\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0$ has no rational roots.

$\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0----(*)$

if n=2 then we have :$x^2+2x+2=0$ no real solution

if m is a solution of original equation then m<0

multiply both sides with n ! we obtain :

$x^n+nx^{n-1}+-----+n! x+ n!=0$

using "the rational zero theorem"

if the original equation has a rational solution m<0 then n! must be a multiple of it(m is a negative integer)

replacing x with any negative factor of n! to (*)will not be zero ,so no rational root exists

in fact :

---+$\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=e^x$

[FONT=細](Maclaurin expasion of $e^x$)[/FONT]

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- Feb 24, 2012

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replacing x with any negative factor of n! to (*)will not be zero

This part is a little vauge to me. To see what I mean, what if my orginal question was to show that there are no

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- Jan 25, 2013

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$\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0----(*)$

if n is even and m (a negative integer ) is a root then (*) becomes

$\displaystyle \frac{m^n}{n!}+\frac{m^{n-2}}{(n-2)!}\cdots+\frac{m^2}{2!}+1$

$- \dfrac{k^{n-1}}{(n-1)!}- \dfrac{k^{n-3}}{(n-3)!}\cdots-\dfrac{k}{1!}$

will not be zero,(the calculation is very tedious)

here k=-m is a positive integer

likewise if n is odd then (*) becomes

$\displaystyle -\frac{k^n}{n!}-\frac{k^{n-2}}{(n-2)!}\cdots-\frac{k^2}{2!}+1$

$+ \dfrac{m^{n-1}}{(n-1)!}+ \dfrac{m^{n-3}}{(n-3)!}\cdots+\dfrac{m}{1!}$

also will not be zero

so there is no integer root for original equation

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