2022 SSMO Team Round Problems/Problem 10

Problem

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

Solution

By Vieta's relation we get, $\sum_{cyc}{}\alpha=2$ $\sum_{cyc}{}\alpha\beta=0$ and $\prod_{cyc}{}\alpha=4=\beta\gamma=\frac{4}{\alpha}$

Therefore we have to find the value of $\prod_{cyc}{}(\alpha^3+\beta\gamma)=\prod_{cyc}{}\left(\alpha^3+\frac{4}{\alpha}\right)=\prod_{cyc}{}\left(\frac{\alpha^4+4}{\alpha}\right)$ $\prod_{cyc}{}\left(\frac{\alpha^4+4}{\alpha}\right)=\frac{\prod_{cyc}{}(\alpha-(1+i))(\alpha-(1-i))(\alpha+(1-i))(\alpha+(1+i))}{\alpha\beta\gamma}$

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2022 SSMO Team Round Problems/Problem 10

Problem

Solution