Question Number 69479 by MJS last updated on 24/Sep/19
![to Sir Aifour: we can construct polynomes of both 3^(rd) and 4^(th) degree in a way that the constants are ∈Z or ∈Q and the solutions are not trivial i.e. (t−α)(t+(α/2)−(√β))(t+(α/2)+(√β))=0∧t=x+(γ/3) ⇔ x^3 +γx^2 −(((3α^2 )/4)+β−(γ^2 /3))x−((α^3 /4)+((α^2 γ)/4)−αβ+((βγ)/3)−(γ^3 /(27)))=0 or the more complicated with sinus/cosinus (x−α−(√β)−(√γ)−(√δ))(x−α−(√β)+(√γ)+(√δ))(x−α+(√β)−(√γ)+(√δ))(x−α+(√β)+(√γ)−(√δ))=0 where all constants ∈Q if (√(βγδ))∈Q I could not find a similar construction for a polynome of 5^(th) degree, where the 5 roots are of comparable complexity [(x−a)(x−b−ci)(x−b+ci)(x−d−ei)(x−d+ei) doesn′t count] maybe you should at first focus on this](Q69479.png)
$$\mathrm{to}\:\mathrm{Sir}\:\mathrm{Aifour}: \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{construct}\:\mathrm{polynomes}\:\mathrm{of}\:\mathrm{both}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{and} \\ $$$$\mathrm{4}^{\mathrm{th}} \:\mathrm{degree}\:\mathrm{in}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{constants}\:\mathrm{are} \\ $$$$\in\mathbb{Z}\:\mathrm{or}\:\in\mathbb{Q}\:\mathrm{and}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{are}\:\mathrm{not}\:\mathrm{trivial} \\ $$$$\mathrm{i}.\mathrm{e}. \\ $$$$\left({t}−\alpha\right)\left({t}+\frac{\alpha}{\mathrm{2}}−\sqrt{\beta}\right)\left({t}+\frac{\alpha}{\mathrm{2}}+\sqrt{\beta}\right)=\mathrm{0}\wedge{t}={x}+\frac{\gamma}{\mathrm{3}} \\ $$$$\Leftrightarrow \\ $$$${x}^{\mathrm{3}} +\gamma{x}^{\mathrm{2}} −\left(\frac{\mathrm{3}\alpha^{\mathrm{2}} }{\mathrm{4}}+\beta−\frac{\gamma^{\mathrm{2}} }{\mathrm{3}}\right){x}−\left(\frac{\alpha^{\mathrm{3}} }{\mathrm{4}}+\frac{\alpha^{\mathrm{2}} \gamma}{\mathrm{4}}−\alpha\beta+\frac{\beta\gamma}{\mathrm{3}}−\frac{\gamma^{\mathrm{3}} }{\mathrm{27}}\right)=\mathrm{0} \\ $$$$\mathrm{or}\:\mathrm{the}\:\mathrm{more}\:\mathrm{complicated}\:\mathrm{with}\:\mathrm{sinus}/\mathrm{cosinus} \\ $$$$ \\ $$$$\left({x}−\alpha−\sqrt{\beta}−\sqrt{\gamma}−\sqrt{\delta}\right)\left({x}−\alpha−\sqrt{\beta}+\sqrt{\gamma}+\sqrt{\delta}\right)\left({x}−\alpha+\sqrt{\beta}−\sqrt{\gamma}+\sqrt{\delta}\right)\left({x}−\alpha+\sqrt{\beta}+\sqrt{\gamma}−\sqrt{\delta}\right)=\mathrm{0} \\ $$$$\mathrm{where}\:\mathrm{all}\:\mathrm{constants}\:\in\mathbb{Q}\:\mathrm{if}\:\sqrt{\beta\gamma\delta}\in\mathbb{Q} \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{could}\:\mathrm{not}\:\mathrm{find}\:\mathrm{a}\:\mathrm{similar}\:\mathrm{construction}\:\mathrm{for} \\ $$$$\mathrm{a}\:\mathrm{polynome}\:\mathrm{of}\:\mathrm{5}^{\mathrm{th}} \:\mathrm{degree},\:\mathrm{where}\:\mathrm{the}\:\mathrm{5}\:\mathrm{roots} \\ $$$$\mathrm{are}\:\mathrm{of}\:\mathrm{comparable}\:\mathrm{complexity} \\ $$$$\left[\left({x}−{a}\right)\left({x}−{b}−{c}\mathrm{i}\right)\left({x}−{b}+{c}\mathrm{i}\right)\left({x}−{d}−{e}\mathrm{i}\right)\left({x}−{d}+{e}\mathrm{i}\right)\right. \\ $$$$\left.\mathrm{doesn}'\mathrm{t}\:\mathrm{count}\right] \\ $$$$\mathrm{maybe}\:\mathrm{you}\:\mathrm{should}\:\mathrm{at}\:\mathrm{first}\:\mathrm{focus}\:\mathrm{on}\:\mathrm{this} \\ $$
Commented by TawaTawa last updated on 24/Sep/19
$$\mathrm{What}\:\mathrm{is}\:\:\alpha,\:\beta,\:\gamma\:\mathrm{and}\:\delta\:\:\mathrm{sir} \\ $$
Commented by TawaTawa last updated on 24/Sep/19
$$\mathrm{I}\:\mathrm{mean}\:\mathrm{the}\:\mathrm{values} \\ $$