Suppose that $a_0, a_1, \ldots a_n \in \mathbb R$ and the polynomial $P(x) = \sum_{k=0}^n a_kx^k$ has all real roots. I'm supposed to show that $$ Q(x) = \sum_{k=0}^n \frac {a_k} {k!}x^k $$ also has this property, i.e. it has all $n$ (possibly non-distinct) real roots.

I know that I'm supposed to show my progress, but I can't find any useful path. Perhaps a proof by induction on $n$ would be helpful - differentiating $Q$ leaves a similar polynomial to work with. I wasn't able to progress though. I would appreciate some clue (not full solution!).

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    $\begingroup$ Did you see this? math.stackexchange.com/questions/338722/… $\endgroup$ – Notsredt Mar 2 '19 at 16:00
  • $\begingroup$ @Nastar no, but it apparently solves my problem. How to mark it? $\endgroup$ – enedil Mar 2 '19 at 16:04
  • $\begingroup$ No idea. Nice question, though. Is your source for this question same as the source given in that answer? $\endgroup$ – Notsredt Mar 2 '19 at 17:44
  • $\begingroup$ @Nastar no, its my homework set from real analysis class (second semester). $\endgroup$ – enedil Mar 2 '19 at 18:07
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    $\begingroup$ I've seen this question before, it was in one of the problem sets Don Zagier gave after his lectures: www-history.mcs.st-and.ac.uk/ems/Zagier/Problems.html $\endgroup$ – Rybin Dmitry Mar 2 '19 at 19:16

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