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Jul 1, 2021 · The given condition can be simplified as $$\beta^n+\gamma^n+\delta^n=(\beta+\gamma+\delta)^n$$ If $\beta,\gamma,\delta$ is non negative then we can use power-mean inequality to get, $${\beta^n+\gamma^n+\delta^n\over3^n}=\left({\beta+\gamma+\delta\over 3}\right)^n\le{\beta^n+\gamma^n+\delta^n\over 3}$$ $$\implies\alpha^n=\beta^n+\gamma^n+\delta^n=0$$ This contradicts $\alpha\ne0$
Aug 5, 2015 · Using the root-coefficient relationship, we have the following: $$\alpha + \beta + \gamma = -1,$$ $$\alpha\beta\gamma = -1,$$ $$\alpha\beta + \beta\gamma + \gamma ...
Jan 25, 2019 · $\alpha ^2+\beta^2+\gamma^2=(\alpha +\beta+\gamma)^2-2(\alpha \beta+\beta \gamma+\alpha \gamma)=4-2(\alpha \beta+\beta \gamma+\alpha \gamma)$ I am not sure how to figure out $2(\alpha \beta+\beta \gamma+\alpha \gamma)$ term.I am not going for long method of finding eigen values by characteristic polynomial since this question came in $2$ marks so I am trying to find out quick way to solve this ...
Dec 19, 2018 · Cases of $\alpha^{'}\geq 5$ included cases of any of the other three ($\beta'\leq6;\gamma'\leq5;\delta'\leq6$) not following together with or putting similar argument for other three conditions, we can contemplate that all in all $4\times 3=12$ terms are already subtracted and that we need to now add the "at least two disobeyed condition cases"($^4C_2=6$ in number).
Jul 11, 2020 · Also $\alpha\beta+\beta\gamma+\gamma\alpha=-\alpha\beta\gamma$ so, yes, I can see why you are asking...$\alpha^3+\beta^3+\gamma^3=3\alpha\beta\gamma$ $\endgroup$ – Martin Hansen Commented Jul 11, 2020 at 15:30
Oct 28, 2020 · Show that $\alpha^2-3(1+\sqrt{10})\alpha+4=0$, and find similar quadratic equations satisfied by $\beta$, $\gamma$ and $\delta$. Unsure how to approach this question. So far I have:
Apr 14, 2019 · I need to show, for ordinals $\alpha, \beta, \gamma, \delta$, that: if $\beta \leq \alpha < \beta + \gamma$ then $\alpha = \beta + \delta$ for some $\delta < \gamma$. I understand this should follow by well-orderedness of ordinals, but I'm not sure of how to phrase it precisely. (I actually need to show it for specific ordinals, but I feel it ...
Oct 12, 2020 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Oct 12, 2019 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Jan 21, 2023 · Compute $\prod e_i^{t_i}-\sum\alpha^2\beta^2\gamma=\sum\alpha\beta\sum\alpha\beta\gamma-\sum\alpha^2\beta^2\gamma=3\sum\alpha^2\beta\gamma\delta.$ Iterate on this new symmetric polynomial (whose exponent is lexicographically smaller), obtaining similarly: $\sum\alpha^2\beta\gamma\delta=e_1e_4.$
