Question Number 135467 by benjo_mathlover last updated on 13/Mar/21
Answered by EDWIN88 last updated on 13/Mar/21
$$\begin{vmatrix}{\alpha\:\:\:\:\:\beta\:\:\:\:\:\:\gamma}\\{\beta\:\:\:\:\:\gamma\:\:\:\:\:\:\alpha}\\{\gamma\:\:\:\:\:\alpha\:\:\:\:\:\beta}\end{vmatrix}=\:\alpha\left(\beta\gamma−\alpha^{\mathrm{2}} \right)−\beta\left(\beta^{\mathrm{2}} −\alpha\gamma\right)+\gamma\left(\alpha\beta−\gamma^{\mathrm{2}} \right) \\ $$$$=\alpha\beta\gamma−\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} +\alpha\beta\gamma+\alpha\beta\gamma−\gamma^{\mathrm{3}} \\ $$$$=\mathrm{3}\alpha\beta\gamma−\left(\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} +\gamma^{\mathrm{3}} \right)…\left(\mathrm{i}\right) \\ $$$$=\:−\mathrm{3b}−\left[\:−\mathrm{a}^{\mathrm{3}} −\mathrm{3b}\:\right] \\ $$$$=\:\mathrm{a}^{\mathrm{3}} \: \\ $$$$\mathrm{consider}\::\:\mathrm{x}^{\mathrm{3}} +\mathrm{ax}^{\mathrm{2}} +\mathrm{b}=\mathrm{0\begin{cases}{\alpha}\\{\beta}\\{\gamma}\end{cases}} \\ $$$$\Rightarrow\alpha^{\mathrm{3}} \:=\:−\mathrm{a}\alpha^{\mathrm{2}} −\mathrm{b} \\ $$$$\Rightarrow\beta^{\mathrm{3}} =\:−\mathrm{a}\beta^{\mathrm{2}} −\mathrm{b} \\ $$$$\Rightarrow\gamma^{\mathrm{3}} =\:−\mathrm{a}\gamma^{\mathrm{2}} −\mathrm{b} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\:+ \\ $$$$:\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} +\gamma^{\mathrm{3}} =−\mathrm{a}\left(\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} +\gamma^{\mathrm{2}} \right)−\mathrm{3b} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\mathrm{a}\left[\left(\alpha+\beta+\gamma\right)^{\mathrm{2}} −\mathrm{2}\left(\alpha\beta+\alpha\gamma+\beta\gamma\right)\right]−\mathrm{3b} \\ $$$$\:\:\:\:=−\mathrm{a}\left[\:\left(−\mathrm{a}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{0}\right)\right]−\mathrm{3b}=−\mathrm{a}^{\mathrm{3}} −\mathrm{3b} \\ $$
Commented by mr W last updated on 13/Mar/21
$$\:\:\:\:=−\mathrm{a}\left[\:\left(−{a}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{0}\right)\right]−\mathrm{3b}=−\mathrm{a}^{\mathrm{3}} −\mathrm{3b} \\ $$$$\Rightarrow{answer}\:{c} \\ $$
Commented by EDWIN88 last updated on 13/Mar/21
$$\mathrm{hahaha} \\ $$