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Question Number 58915 by MJS last updated on 01/May/19

$$\mathrm{reposting}\mathrm{this}:$$$$x^8-8x^7-16x^6+208x^5-152x^4-928x^3+704x^2+1088x-368=0$$$$\mathrm{nobody}\mathrm{wants}\mathrm{to}\mathrm{try}?{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{beautiful}...$$

Commented by Kunal12588 last updated on 01/May/19

$$TerrifyinglyBeautifulsir$$

Commented by Kunal12588 last updated on 01/May/19

Answered by naka3546 last updated on 01/May/19

$$x^8-8x^7-16x^6+208x^5-152x^4-928x^3+704x^2+1088x-368=0$$$$(x^4-4x^3-16x^2+88x-92)(x^4-4x^3-16x^2-8x+4)=0$$

Commented by naka3546 last updated on 01/May/19

$$x^8-8x^7-16x^6+208x^5-152x^4-928x^3+704x^2+1088x-368=0$$$$\Rightarrow x^8-(4x^7+4x^7)-(16x^6-16x^6+16x^6)+(88x^5+64x^5+64x^5-8x^5)$$$$-(92x^4+352x^4-256x^4-32x^4-4x^4)-928x^3+704x^2+1088x-368=0$$$$+(368x^3-1408x^3+128x^3-16x^3)+(1472x^2-704x^2-64x^2)$$$$+(736x+352x)-368=0$$$$\Rightarrow (x^4-4x^3-16x^2+88x-92)(x^4-4x^3-16x^2-8x+4)=0$$

Commented by MJS last updated on 01/May/19

$$\mathrm{great}!\mathrm{can}\mathrm{you}\mathrm{go}\mathrm{on}\mathrm{like}\mathrm{this}?$$

Answered by MJS last updated on 01/May/19

$$x=w+1$$$$w^8-44w^6+438w^4-1292w^2+529=0$$$$w=\pm \sqrt{v}$$$$v^4-44v^3+438v^2-1292v+529=0$$$$v_{1,2}=\alpha \pm \sqrt{\beta }$$$$v_{3,4}=\gamma \pm \sqrt{\delta }$$$$(v-v_1)(v-v_2)(v-v_3)(v-v_4)=v^4-44v^3+438v^2-1292v+529$$$$-2\alpha -2\gamma =-44\Rightarrow \alpha =-\gamma +22$$$${\alpha }^2+4\alpha \gamma -\beta +{\gamma }^2-\delta =438\Rightarrow \beta =-2{\gamma }^2+44\gamma -\delta +46$$$$-2{\alpha }^2\gamma -2\alpha {\gamma }^2+2\alpha \delta +2\beta \gamma =-1292\Rightarrow \delta =\frac{-{\gamma }^3+33{\gamma }^2-219\gamma +323}{\gamma -11}$$$${\alpha }^2{\gamma }^2-{\alpha }^2\delta -\beta {\gamma }^2+\beta \delta =529\Rightarrow$$$$\Rightarrow {\gamma }^6-66{\gamma }^5+1671{\gamma }^4-20284{\gamma }^3+121407{\gamma }^2-339042\gamma +346969=0$$$$\gamma =u+11$$$$u^6-144u^4+6336u^2-82944=0$$$$u=\sqrt{t}$$$$t^3-144t^2+6336t-82944=0$$$$t=s+48$$$$s^3-576s=0\Rightarrow s_1=-24;s_2=0;s_3=24$$$$s=0$$$$t=48$$$$u=4\sqrt{3}$$$$\gamma =11+4\sqrt{3}\delta =96+48\sqrt{3}\sqrt{\delta }=6\sqrt{2}+2\sqrt{6}$$$$\alpha =11-4\sqrt{3}\beta =96-48\sqrt{3}\sqrt{\beta }=6\sqrt{2}-2\sqrt{6}$$$$v_1=11-6\sqrt{2}-4\sqrt{3}+2\sqrt{6}$$$$v_2=11+6\sqrt{2}-4\sqrt{3}-2\sqrt{6}$$$$v_3=11-6\sqrt{2}+4\sqrt{3}-2\sqrt{6}$$$$v_4=11+6\sqrt{2}+4\sqrt{3}+2\sqrt{6}$$$$w_{1,2}=\pm \sqrt{v_1}$$$$w_{3,4}=\pm \sqrt{v_2}$$$$w_{5,6}=\pm \sqrt{v_3}$$$$w_{7,8}=\pm \sqrt{v_4}$$$$\sqrt{v_4}=\sqrt{11+6\sqrt{2}+4\sqrt{3}+2\sqrt{6}}=a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}$$$$11+6\sqrt{2}+4\sqrt{3}+2\sqrt{6}=(a^2+2b^2+3c^2+6d^2)o+2(ab+3cd)\sqrt{2}+2(ac+3bd)\sqrt{3}+2(ad+bc)\sqrt{6}$$$$\mathrm{now}\mathrm{an}\mathrm{obvious}\mathrm{solution}\mathrm{is}a=0\wedge b=c=d=1$$$$\Rightarrow \sqrt{v_4}=\sqrt{11+6\sqrt{2}+4\sqrt{3}+2\sqrt{6}}=\sqrt{2}+\sqrt{3}+\sqrt{6}$$$$\mathrm{similar}$$$$\sqrt{v_1}=\sqrt{2}+\sqrt{3}-\sqrt{6}$$$$\sqrt{v_2}=-\sqrt{2}+\sqrt{3}+\sqrt{6}$$$$\sqrt{v_3}=\sqrt{2}-\sqrt{3}+\sqrt{6}$$$$w_1=-\sqrt{2}-\sqrt{3}+\sqrt{6}\Rightarrow x_1=1-\sqrt{2}-\sqrt{3}+\sqrt{6}$$$$w_2=\sqrt{2}+\sqrt{3}-\sqrt{6}\Rightarrow x_2=1+\sqrt{2}+\sqrt{3}-\sqrt{6}$$$$w_3=\sqrt{2}-\sqrt{3}-\sqrt{6}\Rightarrow x_3=1+\sqrt{2}-\sqrt{3}-\sqrt{6}$$$$w_4=-\sqrt{2}+\sqrt{3}+\sqrt{6}\Rightarrow x_4=1-\sqrt{2}+\sqrt{3}+\sqrt{6}$$$$w_5=-\sqrt{2}+\sqrt{3}-\sqrt{6}\Rightarrow x_5=1-\sqrt{2}+\sqrt{3}-\sqrt{6}$$$$w_6=\sqrt{2}-\sqrt{3}+\sqrt{6}\Rightarrow x_6=1+\sqrt{2}-\sqrt{3}+\sqrt{6}$$$$w_7=-\sqrt{2}-\sqrt{3}-\sqrt{6}\Rightarrow x_7=1-\sqrt{2}-\sqrt{3}-\sqrt{6}$$$$w_8=\sqrt{2}+\sqrt{3}+\sqrt{6}\Rightarrow x_8=1+\sqrt{2}+\sqrt{3}+\sqrt{6}$$

Commented by Kunal12588 last updated on 01/May/19

$$greatsir$$$$x=1\pm \sqrt{2}\sqrt{3}\pm \sqrt{2}\pm \sqrt{3}$$

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