Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 106815 by pticantor last updated on 07/Aug/20

please help me to show that the equation X^n +aX+c=0 can not have more than 3 reals solutions

$$\:\:\:\boldsymbol{{please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}\:\boldsymbol{{to}}\:\boldsymbol{{show}} \\ $$$$\boldsymbol{{that}}\:\:\boldsymbol{{the}}\:\boldsymbol{{equation}}\: \\ $$$$\:\boldsymbol{{X}}^{\boldsymbol{{n}}} +\boldsymbol{{aX}}+\boldsymbol{{c}}=\mathrm{0}\:\boldsymbol{{can}}\:\boldsymbol{{not}}\:\boldsymbol{{have}} \\ $$$$\boldsymbol{{more}}\:\boldsymbol{{than}}\:\mathrm{3}\:\boldsymbol{{reals}}\:\boldsymbol{{solutions}} \\ $$$$ \\ $$

Answered by 1549442205PVT last updated on 07/Aug/20

f(X)=X^n +aX+c⇒f ′(X)=nX^(n−1) +a(1) i)The n odd ⇒n=2k+1 (1)⇔(2k+1)X^(2k) =−a(2) +If a>0 then (2) has no roots ⇒f(X) has one unique root since f(X) is increasing function +If a=0⇒f(X) has only root x=^(2k+1) (√(−c)) +If a<0 then (2) has two roots X=±^(2k) (√((−a)/(2k+1))) ⇒f(X) has no more three roots ii)The n even⇒n=2k.Then (1)⇔2kX^(2k−1) =−a⇔X=^(2k−1) (√((−a)/(2k))) is the unique root ⇒f(X) has no more two roots Thus,in all cases we see that f(X) has no more three solutions (q.e.d) Note:Here we need to understand that the roots of the eqn. X^n +aX+c=0 that we are mentioning be real roots because if not as we were known an arbitrary polynomial of degree n being always has n roots in C

$$\mathrm{f}\left(\mathrm{X}\right)=\mathrm{X}^{\mathrm{n}} +\mathrm{aX}+\mathrm{c}\Rightarrow\mathrm{f}\:'\left(\mathrm{X}\right)=\mathrm{nX}^{\mathrm{n}−\mathrm{1}} +\mathrm{a}\left(\mathrm{1}\right) \\ $$$$\left.\mathrm{i}\right)\mathrm{The}\:\mathrm{n}\:\mathrm{odd}\:\Rightarrow\mathrm{n}=\mathrm{2k}+\mathrm{1} \\ $$$$\left(\mathrm{1}\right)\Leftrightarrow\left(\mathrm{2k}+\mathrm{1}\right)\mathrm{X}^{\mathrm{2k}} =−\mathrm{a}\left(\mathrm{2}\right) \\ $$$$+\mathrm{If}\:\mathrm{a}>\mathrm{0}\:\mathrm{then}\:\left(\mathrm{2}\right)\:\mathrm{has}\:\mathrm{no}\:\mathrm{roots} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{X}\right)\:\mathrm{has}\:\mathrm{one}\:\mathrm{unique}\:\mathrm{root}\:\mathrm{since}\:\mathrm{f}\left(\mathrm{X}\right) \\ $$$$\mathrm{is}\:\mathrm{increasing}\:\mathrm{function} \\ $$$$+\mathrm{If}\:\mathrm{a}=\mathrm{0}\Rightarrow\mathrm{f}\left(\mathrm{X}\right)\:\mathrm{has}\:\mathrm{only}\:\mathrm{root}\:\mathrm{x}=^{\mathrm{2k}+\mathrm{1}} \sqrt{−\mathrm{c}} \\ $$$$+\mathrm{If}\:\mathrm{a}<\mathrm{0}\:\mathrm{then}\:\left(\mathrm{2}\right)\:\mathrm{has}\:\mathrm{two}\:\mathrm{roots} \\ $$$$\mathrm{X}=\pm\:^{\mathrm{2k}} \sqrt{\frac{−\mathrm{a}}{\mathrm{2k}+\mathrm{1}}} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{X}\right)\:\mathrm{has}\:\mathrm{no}\:\mathrm{more}\:\mathrm{three}\:\mathrm{roots} \\ $$$$\left.\mathrm{ii}\right)\mathrm{The}\:\mathrm{n}\:\mathrm{even}\Rightarrow\mathrm{n}=\mathrm{2k}.\mathrm{Then} \\ $$$$\left(\mathrm{1}\right)\Leftrightarrow\mathrm{2kX}^{\mathrm{2k}−\mathrm{1}} =−\mathrm{a}\Leftrightarrow\mathrm{X}=\:^{\mathrm{2k}−\mathrm{1}} \sqrt{\frac{−\mathrm{a}}{\mathrm{2k}}} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{unique}\:\mathrm{root} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{X}\right)\:\mathrm{has}\:\mathrm{no}\:\mathrm{more}\:\mathrm{two}\:\mathrm{roots} \\ $$$$\mathrm{Thus},\mathrm{in}\:\mathrm{all}\:\mathrm{cases}\:\mathrm{we}\:\mathrm{see}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{X}\right)\:\mathrm{has} \\ $$$$\mathrm{no}\:\mathrm{more}\:\mathrm{three}\:\mathrm{solutions}\:\left(\mathrm{q}.\mathrm{e}.\mathrm{d}\right) \\ $$$$\boldsymbol{\mathrm{Note}}:\boldsymbol{\mathrm{Here}}\:\boldsymbol{\mathrm{we}}\:\boldsymbol{\mathrm{need}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{understand}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{roots}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{eqn}}.\:\boldsymbol{\mathrm{X}}^{\boldsymbol{\mathrm{n}}} +\boldsymbol{\mathrm{aX}}+\boldsymbol{\mathrm{c}}=\mathrm{0}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{we}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{mentioning}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{roots}} \\ $$$$\boldsymbol{\mathrm{because}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{not}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{we}}\:\boldsymbol{\mathrm{were}}\:\boldsymbol{\mathrm{known}} \\ $$$$\mathrm{a}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{arbitrary}}\:\boldsymbol{\mathrm{polynomial}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{degree}}\:\boldsymbol{\mathrm{n}}\: \\ $$$$\boldsymbol{\mathrm{being}}\:\boldsymbol{\mathrm{always}}\:\boldsymbol{\mathrm{has}}\:\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{roots}}\:\boldsymbol{\mathrm{in}}\:\mathbb{C} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com