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Question Number 50445 by ANTARES VY last updated on 16/Dec/18

The roots of the fallowing functions are the Sequences of arithmetic progressiyon f(x)=x^5 −20x^4 +ax^3 +bx^2 +cx+24 f(8)=?

$$\boldsymbol{\mathrm{The}}\:\:\boldsymbol{\mathrm{roots}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{fallowing}} \\ $$$$\boldsymbol{\mathrm{functions}}\:\:\boldsymbol{\mathrm{are}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{Sequences}}\:\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{arithmetic}}\:\:\boldsymbol{\mathrm{progressiyon}} \\ $$$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathrm{x}}^{\mathrm{5}} −\mathrm{20}\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\boldsymbol{\mathrm{ax}}^{\mathrm{3}} +\boldsymbol{\mathrm{bx}}^{\mathrm{2}} +\boldsymbol{\mathrm{cx}}+\mathrm{24} \\ $$$$\boldsymbol{\mathrm{f}}\left(\mathrm{8}\right)=? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 16/Dec/18

pls check the question.. if f(x)=0 value of x is in terms of a,b,c so f(8)=8^5 −20×8^4 +a×8^3 +b×8^2 +c×8+24

$${pls}\:{check}\:{the}\:{question}.. \\ $$$${if}\:{f}\left({x}\right)=\mathrm{0}\: \\ $$$${value}\:{of}\:{x}\:{is}\:{in}\:{terms}\:{of}\:{a},{b},{c} \\ $$$${so}\: \\ $$$${f}\left(\mathrm{8}\right)=\mathrm{8}^{\mathrm{5}} −\mathrm{20}×\mathrm{8}^{\mathrm{4}} +{a}×\mathrm{8}^{\mathrm{3}} +{b}×\mathrm{8}^{\mathrm{2}} +{c}×\mathrm{8}+\mathrm{24} \\ $$

Answered by ajfour last updated on 16/Dec/18

let roots be x_i = p−2q, p−q, p, p+q, p+2q ⇒ Σx_i = 5p = 20 ⇒ p = 4 Πx_i = p(p^2 −q^2 )(p^2 −4q^2 )= −24 let q^2 = s ⇒ (16−s)(16−4s)+6 = 0 4s^2 −80s+262 = 0 or 2s^2 −40s+131 = 0 s = q^2 = 10±((√(138))/2) (not a convinient value..) a = Σx_1 x_2 b = −Σx_1 x_2 x_3 c = Σx_1 x_2 x_3 x_4 f(8) = 8^5 −20(8)^4 +a(8)^3 +b(8)^2 +8c+24

$${let}\:{roots}\:{be} \\ $$$${x}_{{i}} \:=\:{p}−\mathrm{2}{q},\:{p}−{q},\:{p},\:{p}+{q},\:{p}+\mathrm{2}{q} \\ $$$$\Rightarrow\:\Sigma{x}_{{i}} =\:\mathrm{5}{p}\:=\:\mathrm{20}\:\:\Rightarrow\:\:{p}\:=\:\mathrm{4} \\ $$$$\Pi{x}_{{i}} =\:{p}\left({p}^{\mathrm{2}} −{q}^{\mathrm{2}} \right)\left({p}^{\mathrm{2}} −\mathrm{4}{q}^{\mathrm{2}} \right)=\:−\mathrm{24} \\ $$$${let}\:{q}^{\mathrm{2}} \:=\:{s} \\ $$$$\Rightarrow\:\:\left(\mathrm{16}−{s}\right)\left(\mathrm{16}−\mathrm{4}{s}\right)+\mathrm{6}\:=\:\mathrm{0} \\ $$$$\mathrm{4}{s}^{\mathrm{2}} −\mathrm{80}{s}+\mathrm{262}\:=\:\mathrm{0} \\ $$$${or}\:\:\:\mathrm{2}{s}^{\mathrm{2}} −\mathrm{40}{s}+\mathrm{131}\:=\:\mathrm{0} \\ $$$${s}\:=\:{q}^{\mathrm{2}} \:=\:\mathrm{10}\pm\frac{\sqrt{\mathrm{138}}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\left({not}\:{a}\:{convinient}\:{value}..\right) \\ $$$$\:\:\:\:{a}\:=\:\Sigma{x}_{\mathrm{1}} {x}_{\mathrm{2}} \\ $$$$\:\:\:\:{b}\:=\:−\Sigma{x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} \\ $$$$\:\:\:\:{c}\:=\:\Sigma{x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} {x}_{\mathrm{4}} \\ $$$${f}\left(\mathrm{8}\right)\:=\:\mathrm{8}^{\mathrm{5}} −\mathrm{20}\left(\mathrm{8}\right)^{\mathrm{4}} +{a}\left(\mathrm{8}\right)^{\mathrm{3}} +{b}\left(\mathrm{8}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{8}{c}+\mathrm{24}\: \\ $$

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