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Question Number 93414 by abdomathmax last updated on 13/May/20
![let p(x)=((x^n (4−2x)^n )/(n!)) 1) prove that p^((k)) (0)=p^((k)) (2)=0 for all k∈[1,n−1] 2) prove that ∀m∈N p^((m)) (0) and p^((m)) (2) are integrs](Q93414.png)
$${let}\:{p}\left({x}\right)=\frac{{x}^{{n}} \left(\mathrm{4}−\mathrm{2}{x}\right)^{{n}} }{{n}!} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:{p}^{\left({k}\right)} \left(\mathrm{0}\right)={p}^{\left({k}\right)} \left(\mathrm{2}\right)=\mathrm{0}\:{for}\:{all}\:{k}\in\left[\mathrm{1},{n}−\mathrm{1}\right] \\ $$$$\left.\mathrm{2}\right)\:\:{prove}\:{that}\:\:\forall{m}\in{N}\:\:\:\:{p}^{\left({m}\right)} \left(\mathrm{0}\right)\:{and}\:{p}^{\left({m}\right)} \left(\mathrm{2}\right)\:{are}\:{integrs} \\ $$