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GeometryQuestion and Answers: Page 78

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Find the remainder when x + x^(25) + x^(49) + x^(81) is divided by x^3 − 1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\:\:\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{25}} \:+\:\mathrm{x}^{\mathrm{49}} \:+\:\mathrm{x}^{\mathrm{81}} \:\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\:\mathrm{x}^{\mathrm{3}} \:−\:\mathrm{1} \\ $$

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In a AB^△ C: { ((a+b+c=2(h_a +h_b +h_c ))),((a^2 +b^2 +c^2 =6abc)),((h_a ^2 +h_b ^2 +h_c ^2 =6h_a .h_b .h_c )) :} find:∡A

$$\boldsymbol{\mathrm{In}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{A}}\overset{\bigtriangleup} {\boldsymbol{\mathrm{B}C}}: \\ $$$$\begin{cases}{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}=\mathrm{2}\left(\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{a}}} +\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{b}}} +\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{c}}} \right)}\\{\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} =\mathrm{6}\boldsymbol{\mathrm{abc}}}\\{\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{2}} +\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{b}}} ^{\mathrm{2}} +\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{2}} =\mathrm{6}\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{a}}} .\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{b}}} .\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{c}}} }\end{cases} \\ $$$$\boldsymbol{\mathrm{find}}:\measuredangle\boldsymbol{\mathrm{A}} \\ $$

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