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Category: Algebra

A-polynomial-f-x-with-rational-coefficients-leaves-remainder-15-when-divided-by-x-3-and-remainder-2x-1-when-divided-by-x-1-2-Find-the-remainder-when-f-x-is-divided-by-x-3-x-1-2-

Question Number 19101 by Tinkutara last updated on 04/Aug/17 $$\mathrm{A}\:\mathrm{polynomial}\:{f}\left({x}\right)\:\mathrm{with}\:\mathrm{rational} \\ $$$$\mathrm{coefficients}\:\mathrm{leaves}\:\mathrm{remainder}\:\mathrm{15},\:\mathrm{when} \\ $$$$\mathrm{divided}\:\mathrm{by}\:{x}\:−\:\mathrm{3}\:\mathrm{and}\:\mathrm{remainder}\:\mathrm{2}{x}\:+\:\mathrm{1}, \\ $$$$\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} .\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remainder}\:\mathrm{when}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left({x}\:−\:\mathrm{3}\right)\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} . \\ $$ Commented…

Find-in-closed-form-n-N-0-1-ln-1-x-2-ln-n-1-x-dx-

Question Number 150121 by mathdanisur last updated on 09/Aug/21 $$\mathrm{Find}\:\mathrm{in}\:\mathrm{closed}\:\mathrm{form}:\:\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{ln}\left(\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} \right)\mathrm{ln}^{\boldsymbol{\mathrm{n}}} \left(\mathrm{1}\:-\:\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$ Answered by Kamel last updated on 09/Aug/21…

Question-150097

Question Number 150097 by mathdanisur last updated on 09/Aug/21 Answered by Kamel last updated on 09/Aug/21 $${u}_{{n}} =\underset{{p}=\mathrm{0}} {\overset{{m}} {\prod}}\gamma_{{n}+{p}} =\left(\gamma+\frac{\mathrm{1}}{\mathrm{2}{n}}\right)^{{m}+\mathrm{1}} +{o}\left(\frac{\mathrm{1}}{{n}}\right)=\gamma^{{m}+\mathrm{1}} +\left({m}+\mathrm{1}\right)\frac{\gamma^{{m}} }{\mathrm{2}{n}}+{o}\left(\frac{\mathrm{1}}{{n}}\right) \\…

x-dx-x-8-1-

Question Number 150095 by mathdanisur last updated on 09/Aug/21 $$\Omega\:=\int\:\frac{\mathrm{x}\:\mathrm{dx}}{\mathrm{x}^{\mathrm{8}} \:-\:\mathrm{1}}\:=\:? \\ $$ Answered by Ar Brandon last updated on 09/Aug/21 $$\Omega=\int\frac{{xdx}}{{x}^{\mathrm{8}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{d}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \right)^{\mathrm{4}}…