Question Number 57947 by maxmathsup by imad last updated on 14/Apr/19
![let P(x)=(1+ix)^n −1−ni with x real and n integr natural 1) find the roots of P(x) 2) factorize P(x) inside C[x] 3) factorize P(x) inside R[x] 4) decompose the fraction F(x) =((P^((1)) (x))/(P(x))) inside C(x) P^((1)) is the derivative of P .](Q57947.png)
$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{{n}} −\mathrm{1}−{ni}\:\:\:\:{with}\:{x}\:{real}\:{and}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{P}\left({x}\right)\:{inside}\:{R}\left[{x}\right] \\ $$$$\left.\mathrm{4}\right)\:{decompose}\:{the}\:{fraction}\:{F}\left({x}\right)\:=\frac{{P}^{\left(\mathrm{1}\right)} \left({x}\right)}{{P}\left({x}\right)}\:{inside}\:{C}\left({x}\right) \\ $$$${P}^{\left(\mathrm{1}\right)} \:{is}\:{the}\:{derivative}\:{of}\:{P}\:. \\ $$