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If $f(x)= \frac{x^n}{n!} +\frac{x^{(n-1)}}{(n−1)!}+···+x+1$,then show that $f(x)=0$ has no repeated roots.

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Do you have any thoughts on the problem? – Jonas Meyer Dec 15 '12 at 7:34

On the contrary if $\alpha$ was a repeated root, then $f(\alpha) = f'(\alpha) = 0$

Subtracting the above two equations tells you something about $\alpha$. Now can such an alpha be a root?

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Consider that the derivative of a polynomial is zero at a root of that polynomial if that root is repeated. Now use induction on $n$.

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