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-t-2-t-2-0-t-x-

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

\in Z or \in Q and the solutions are not trivial

i . e .

(t - α) (t + \frac{α}{2} - \sqrt{β}) (t + \frac{α}{2} + \sqrt{β}) = 0 \land t = x + \frac{γ}{3}

\Leftrightarrow

x^{3} + γ x^{2} - (\frac{3 α^{2}}{4} + β - \frac{γ^{2}}{3}) x - (\frac{α^{3}}{4} + \frac{α^{2} γ}{4} - α β + \frac{β γ}{3} - \frac{γ^{3}}{27}) = 0

or the more complicated with sinus / cosinus

(x - α - \sqrt{β} - \sqrt{γ} - \sqrt{δ}) (x - α - \sqrt{β} + \sqrt{γ} + \sqrt{δ}) (x - α + \sqrt{β} - \sqrt{γ} + \sqrt{δ}) (x - α + \sqrt{β} + \sqrt{γ} - \sqrt{δ}) = 0

where all constants \in Q if \sqrt{β γ δ} \in 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 - c i) (x - b + c i) (x - d - e i) (x - d + e i)

{doesn}^{'} t count]

maybe you should at first focus on this

Commented by TawaTawa last updated on 24/Sep/19

What is α, β, γ and δ sir

Commented by TawaTawa last updated on 24/Sep/19

I mean the values