Question Number 176898 by adhigenz last updated on 27/Sep/22
Answered by Rasheed.Sindhi last updated on 27/Sep/22
$$\alpha+\beta=−{m}\:,\:\alpha\beta={n} \\ $$$$\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} =\left(\alpha+\beta\right)\left(\:\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{3}\alpha\beta\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=−{m}\left(\:\left(−{m}\right)^{\mathrm{2}} −\mathrm{3}\left({n}\right)\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=−{m}^{\mathrm{3}} +\mathrm{3}{n} \\ $$$$\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} \leqslant\mathrm{4}\Rightarrow−{m}^{\mathrm{3}} +\mathrm{3}{n}\leqslant\mathrm{4} \\ $$$${n}\leqslant\frac{\mathrm{4}+{m}^{\mathrm{3}} }{\mathrm{3}} \\ $$$$…. \\ $$$$… \\ $$