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| | ==Solution== | | ==Solution== |
| − | By Vieta's relation we get, <math>\sum_{cyc}{}\alpha=2</math> <math>\sum_{cyc}{}\alpha\beta=0</math> and <math>\prod_{cyc}{}\alpha=4</math>
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| − | Therefore we have to find the value of <cmath>\prod_{cyc}{}(\alpha^3+\beta\gamma)\implies\prod_{cyc}{}\left(\alpha^3+\frac{4}{\alpha}\right)\implies\prod_{cyc}{}\left(\frac{\alpha^4+4}{\alpha}\right)</cmath>
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| − | <cmath>\prod_{cyc}{}\left(\frac{\alpha^4+4}{\alpha}\right)=\frac{\prod_{cyc}{}(\alpha-(1+i))(\alpha-(1-i))(\alpha+(1-i))(\alpha+(1+i))}{\alpha\beta\gamma}</cmath>
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Latest revision as of 14:33, 9 September 2025
are the roots of the polynomial 
, find ![\[(\alpha^3+\beta\gamma)(\beta^3+\gamma\alpha)(\gamma^3+\alpha\beta).\]](http://latex.artofproblemsolving.com/a/4/6/a463ccc3de2f3710c58da87d8e3465af873f82ad.png)
