Question Number 52449 by maxmathsup by imad last updated on 07/Jan/19
![let j=e^((i2π)/3) and P(x)=(1+jx)^n −(1−jx)^n with n integr natural 1) find roots of P(x) 2)factorize P(x) inside C[x] 3) calculate ∫_0 ^1 P(x)dx. 4) decompose inside C(x) the fraction F(x)=(1/(P(x)))](Q52449.png)
$${let}\:{j}={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:{P}\left({x}\right)=\left(\mathrm{1}+{jx}\right)^{{n}} −\left(\mathrm{1}−{jx}\right)^{{n}} \:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right)\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {P}\left({x}\right){dx}. \\ $$$$\left.\mathrm{4}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{P}\left({x}\right)} \\ $$