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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-




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 .
$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{{n}} −\mathrm{1}−{ni}\:\:\:\:{with}\:{x}\:{real}\:{and}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{P}\left({x}\right)\:{inside}\:{R}\left[{x}\right] \\ $$$$\left.\mathrm{4}\right)\:{decompose}\:{the}\:{fraction}\:{F}\left({x}\right)\:=\frac{{P}^{\left(\mathrm{1}\right)} \left({x}\right)}{{P}\left({x}\right)}\:{inside}\:{C}\left({x}\right) \\ $$$${P}^{\left(\mathrm{1}\right)} \:{is}\:{the}\:{derivative}\:{of}\:{P}\:. \\ $$

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