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Question Number 178476 by Shrinava last updated on 17/Oct/22

$$\mathrm{Find}\mathrm{a}\in \mathbb{R}$$$$\mathrm{Such}\mathrm{that}{\mathrm{x}}_1^{16}+{\mathrm{x}}_2^{16}+{\mathrm{x}}_3^{16}=30$$$$\mathrm{Where}{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3-\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}$$$$\mathrm{equation}:{\mathrm{x}}^3+\mathrm{ax}+1=0$$

Answered by Frix last updated on 17/Oct/22

$$x^3+ax+1=0$$$$\Rightarrow$$$$x_k=\frac{2\sqrt{-3a}}{3}\sin \frac{2\pi k+{\sin }^{-1}\frac{3\sqrt{3}}{2{(-a)}^{3/2}}}{3}\mathrm{with}k=1,2,3$$$$\mathrm{let}\alpha =\frac{3\sqrt{3}}{2{(-a)}^{3/2}}$$$$x_k=\frac{2\sqrt{-3a}}{3}\sin \frac{2\pi k+{\sin }^{-1}\alpha }{3}$$$$x_k^{16}=\frac{65536a^8}{6561}{\sin }^{16}\frac{2\pi k+{\sin }^{-1}\alpha }{3}$$$$\sin \frac{2\pi +{\sin }^{-1}\alpha }{3}=\frac{\sqrt{3}\cos \frac{{\sin }^{-1}\alpha }{3}-\sin \frac{{\sin }^{-1}\alpha }{3}}{2}=\frac{\sqrt{3}c-s}{2}$$$$\sin \frac{4\pi +{\sin }^{-1}\alpha }{3}=-\frac{\sqrt{3}\cos \frac{{\sin }^{-1}\alpha }{3}+\sin \frac{{\sin }^{-1}\alpha }{3}}{2}=-\frac{\sqrt{3}c+s}{2}$$$$\sin \frac{6\pi +{\sin }^{-1}\alpha }{3}=\sin \frac{{\sin }^{-1}\alpha }{3}=s$$$${\left(\frac{\sqrt{3}c-s}{2}\right)}^{16}+{(-\frac{\sqrt{3}c+s}{2})}^{16}+s^{16}=$$$$(\mathrm{with}c=\sqrt{1-s^2})$$$$=\frac{45s^{12}}{2}-\frac{135s^{10}}{2}+\frac{1215s^8}{16}-\frac{1701s^6}{64}-\frac{5103s^4}{512}+\frac{6561s^2}{1024}+\frac{6561}{32768}=$$$$\left(\mathrm{after}\mathrm{some}\mathrm{work}\right)$$$$=\frac{360\cos \left(12\times \frac{{\sin }^{-1}\alpha }{3}\right)-13104\cos \left(6\times \frac{{\sin }^{-1}\alpha }{3}\right)+19305}{32768}=$$$$=\frac{45{\alpha }^4}{512}+\frac{729{\alpha }^2}{1024}+\frac{6561}{32768}=$$$$=\frac{6561}{32768}-\frac{19683}{4086a^3}+\frac{32805}{8192a^6}$$$$\mathrm{multiplied}\mathrm{with}\frac{65536a^8}{6561}$$$$x_1^{16}+x_2^{16}+x_3^{16}=2a^8-48a^5+40a^2=30$$$$\mathrm{we}\mathrm{need}\mathrm{to}\mathrm{solve}$$$$a^8-24a^5+20a^2-15=0$$$$\mathrm{I}\mathrm{can}\mathrm{only}\mathrm{approximate}$$$$a_1\approx -.718638134$$$$a_2\approx 2.85282288$$$$\mathrm{no}\mathrm{other}\mathrm{real}\mathrm{solutions}$$

Commented by Shrinava last updated on 17/Oct/22

$$\mathrm{Sorry}\mathrm{professor},{\mathrm{x}}_1^{16}+{\mathrm{x}}_2^{16}+{\mathrm{x}}_3^{16}=90$$

Commented by Ar Brandon last updated on 17/Oct/22

Greetings, old man �� I already began to feel your absence and was missing you already, dear Sir�� Little did I know that...��

Commented by Frix last updated on 17/Oct/22

$$x_1^{16}+x_2^{16}+x_3^{16}=2a^8-48a^5+40a^2=90$$$$a^8-24a^5+20a^2-45=0$$$$a_1=-1$$$$a_2\approx 2.85944202$$$$\mathrm{no}\mathrm{other}\mathrm{real}\mathrm{solutions}$$

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