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let-P-x-x-5-209x-56-Prove-that-there-exist-two-roots-a-b-such-as-ab-1-Find-out-their-sum-a-b-and-deduce-the-decomposition-of-P-x-in-prime-factors-

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Question Number 78489 by ~blr237~ last updated on 18/Jan/20
let P(x)= x^5 −209x+56 Prove that there exist two roots a,b such as ab=1 Find out their sum ( a+b=?) and deduce the decomposition of P(x) in prime factors.
letP(x)=x5−209x+56Provethatthereexisttworootsa,bsuchasab=1Findouttheirsum(a+b=?)anddeducethedecompositionofP(x)inprimefactors.
Answered by MJS last updated on 18/Jan/20
2 roots with ab=1 ⇒ we have a square factor (x−a)(x−(1/a))=x^2 −(a+(1/a))x+1= [let a+(1/a)=A] =x^2 −Ax+1 ⇒ the other factor is x^3 +αx^2 +βx+56 x^5 −209x+56=(x^2 −Ax+1)(x^3 +αx^2 +βx+56) x^5 −209x+56=x^5 +(α−A)x^4 −(αA−β−1)x^3 −(βA−α−56)x^2 +(β−56A)x+56 ⇒ (1) α−A=0 (2) αA−β−1=0 (3) βA−α−56=0 (4) β−56A+209=0 ⇒ A=4 ⇒ a=2±(√3) ⇒ b=2∓(√3) α=4 β=15 ⇒ x^5 −209x+56= =(x−2−(√3))(x−2+(√3))(x^3 +4x^2 +15x+56) and we need Cardano′s method for the 2^(nd) factor
2rootswithab=1⇒wehaveasquarefactor(x−a)(x−1a)=x2−(a+1a)x+1=[leta+1a=A]=x2−Ax+1⇒theotherfactorisx3+αx2+βx+56x5−209x+56=(x2−Ax+1)(x3+αx2+βx+56)x5−209x+56=x5+(α−A)x4−(αA−β−1)x3−(βA−α−56)x2+(β−56A)x+56⇒(1)α−A=0(2)αA−β−1=0(3)βA−α−56=0(4)β−56A+209=0⇒A=4⇒a=2±3⇒b=2∓3α=4β=15⇒x5−209x+56==(x−2−3)(x−2+3)(x3+4x2+15x+56)andweneedCardano′smethodforthe2ndfactor
Commented by ~blr237~ last updated on 18/Jan/20
thanks sir for this part , but before reaching here we ought to prove the existence of a and b
thankssirforthispart,butbeforereachinghereweoughttoprovetheexistenceofaandb
Commented by MJS last updated on 18/Jan/20
I have no idea how to prove the existence of the roots without calculating them
Ihavenoideahowtoprovetheexistenceoftherootswithoutcalculatingthem
Commented by mr W last updated on 18/Jan/20
f′(x)=5x^4 −209=0 ⇒x_(1,2) =∓(√(√(209/5))) f(x_1 )>0, f(x_2 )<0 f(→−∞)→−∞, f(→+∞)→+∞ ⇒three real roots exist. one <−(√(√(209/5))) one >(√(√(209/5))) and the third one between.
f′(x)=5x4−209=0⇒x1,2=∓209/5f(x1)>0,f(x2)<0f(→−∞)→−∞,f(→+∞)→+∞⇒threerealrootsexist.one<−209/5one>209/5andthethirdonebetween.
Commented by mr W last updated on 18/Jan/20
Commented by mr W last updated on 18/Jan/20
that f(x) has one factor x^2 +px+1 is the proof that two roots exist whose product is 1.
thatf(x)hasonefactorx2+px+1istheproofthattworootsexistwhoseproductis1.
Commented by mr W last updated on 18/Jan/20
solution of MJS sir is absolutely correct and nice.
solutionofMJSsirisabsolutelycorrectandnice.

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