Difference between revisions of "2022 SSMO Team Round Problems/Problem 10"

Latest revision as of 14:33, 9 September 2025

If $\alpha, \beta, \gamma$ are the roots of the polynomial $x^3-2x^2-4$ , find $(\alpha^3+\beta\gamma)(\beta^3+\gamma\alpha)(\gamma^3+\alpha\beta).$

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 ==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>
-Therefore we have to find the value of <cmath>\prod_{cyc}{}(\alpha^3+\beta\gamma)=\prod_{cyc}{}\left(\alpha^3+\frac{4}{\alpha}\right)=\prod_{cyc}{}\left(\frac{\alpha^4+4}{\alpha}\right)</cmath>
-<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>