Question Number 66302 by ajfour last updated on 12/Aug/19
$${solved}\:{the}\:{general}\:{quintic}, \\ $$$${despite}\:{whatever}\:{proof}\:{that}\:{it} \\ $$$${cant}\:{be}\:{solved}\:{in}\:{a}\:{simple}\:{way}! \\ $$
Commented by mr W last updated on 12/Aug/19
$${i}\:{knew}\:{you}'{ll}\:{tell}\:{us}\:{the}\:{good}\:{news}. \\ $$$${thanks}\:{sir}! \\ $$
Commented by ajfour last updated on 12/Aug/19
$${i}\:{haven}'{t}\:{verified}\:{it}\:{yet},\:{but}\:{i} \\ $$$${think}\:{the}\:{method}\:{is}\:{too}\:{lengthy} \\ $$$${but}\:{plausible}! \\ $$$${Sir}\:{consider} \\ $$$$\:{t}^{\mathrm{5}} +{at}^{\mathrm{3}} +{bt}^{\mathrm{2}} +{ct}+{d}=\mathrm{0} \\ $$$${can}\:{one}\:{root}\:{be}\:{independent}\:{of} \\ $$$${the}\:{constant}\:{term}\:{d}\:? \\ $$
Commented by MJS last updated on 12/Aug/19
$$\mathrm{you}\:\mathrm{know}\:\mathrm{that} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}−\alpha_{{i}} \right) \\ $$$$\mathrm{leads}\:\mathrm{to}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{term} \\ $$$$\left(−\mathrm{1}\right)^{{n}} \underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\alpha_{{i}} \\ $$$$\Rightarrow\:\mathrm{the}\:\mathrm{constant}\:\mathrm{factor}\:\mathrm{depends}\:\mathrm{on}\:\mathrm{the}\:\mathrm{roots} \\ $$$$\mathrm{and}\:\mathrm{vice}\:\mathrm{versa}.\:\mathrm{this}\:\mathrm{is}\:\mathrm{obvious},\:\mathrm{or}...? \\ $$