# $\sum_{k=0}^n a_kx^k$ splits $\Rightarrow \sum_{k=0}^n \frac {a_k} {k!}x^k$ splits over reals

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!).

• Did you see this? math.stackexchange.com/questions/338722/… – Notsredt Mar 2 '19 at 16:00
• @Nastar no, but it apparently solves my problem. How to mark it? – enedil Mar 2 '19 at 16:04
• No idea. Nice question, though. Is your source for this question same as the source given in that answer? – Notsredt Mar 2 '19 at 17:44
• @Nastar no, its my homework set from real analysis class (second semester). – enedil Mar 2 '19 at 18:07
• 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 – Rybin Dmitry Mar 2 '19 at 19:16