Question Number 205051 by mr W last updated on 06/Mar/24
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{k}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{expr}{e}\mathrm{ssion}\:\mathrm{x}^{\mathrm{3}} +\:\mathrm{kx}^{\mathrm{2}} −\mathrm{7x}+\mathrm{6}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{re}{s}\mathrm{olved}\:\mathrm{into}\:\mathrm{three}\:\mathrm{linear}\:\mathrm{real}\:\mathrm{factors}. \\ $$
Answered by Frix last updated on 06/Mar/24
$${k}<{c}\:\mathrm{with}\:{c}\:\mathrm{being}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solution}\:\mathrm{of} \\ $$$${c}^{\mathrm{3}} −\frac{\mathrm{49}{c}^{\mathrm{2}} }{\mathrm{24}}+\frac{\mathrm{63}{c}}{\mathrm{2}}−\frac{\mathrm{50}}{\mathrm{3}}=\mathrm{0} \\ $$$${c}\approx.\mathrm{543134153288} \\ $$
Commented by mr W last updated on 06/Mar/24
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Answered by mr W last updated on 07/Mar/24
$${f}\left({x}\right)={x}^{\mathrm{3}} +{kx}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{6} \\ $$$${f}'\left({x}\right)=\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{kx}−\mathrm{7} \\ $$$${at}\:{x}={p}:\:{f}\left({p}\right)=\mathrm{0},\:{f}'\left({p}\right)=\mathrm{0} \\ $$$${p}^{\mathrm{3}} +{kp}^{\mathrm{2}} −\mathrm{7}{p}+\mathrm{6}=\mathrm{0}\:\:\:…\left({i}\right) \\ $$$$\mathrm{3}{p}^{\mathrm{2}} +\mathrm{2}{kp}−\mathrm{7}=\mathrm{0}\:\:\:…\left({ii}\right) \\ $$$${p}^{\mathrm{3}} +\mathrm{7}{p}−\mathrm{12}=\mathrm{0} \\ $$$${p}=\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3945}}}{\mathrm{9}}+\mathrm{6}}−\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3945}}}{\mathrm{9}}−\mathrm{6}} \\ $$$${k}=\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{7}}{{p}}−\mathrm{3}{p}\right)\approx\mathrm{0}.\mathrm{54313415} \\ $$