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.
Show that $f(x)$ has no repeated roots.
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algebra-precalculus
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1Do you have any thoughts on the problem? – 2012-12-15
2 Answers
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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 $n$.