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Question Number 104771 by mathmax by abdo last updated on 23/Jul/20
let f(x) =x^3 +x−3 1) prove that f have one root real α_0 and α_0 ∈ ]1,2[ 2) factorize f(x) inside R[x] and C[x] 3 ) find ∫ (dx/(f(x)))
letf(x)=x3+x−31)provethatfhaveonerootrealα0andα0∈]1,2[2)factorizef(x)insideR[x]andC[x]3)find∫dxf(x)
Answered by abdomathmax last updated on 24/Jul/20
1) f continue and f^′ (x)=3x^2 +1>0 ⇒f is increazing we have f(1) =−1 and f(2) =8+2−3 =7 ⇒ f(1).f(2)<0 ⇒∃α_0 ∈]1,2[ unique /f(α_0 )=0 2)f(x) =(x−α_0 )(x^2 +ax +b) ⇒ x^3 +ax^2 +bx−α_0 x^2 −α_o a x−bα_0 =x^3 +x−3 ⇒ (a−α_0 )x^2 +(b−α_o a)x −bα_0 =x−3 ⇒ { ((a−α_0 =0)),((b−α_0 a =1 and −bα_0 =−3 ⇒)) :} a =α_0 and b =(3/α_0 ) f(x) =(x−α_0 )(x^2 +α_0 x +(3/α_0 )) this is tbe factoruzation at R[x] x^2 +α_0 x +(3/α_0 ) =0 →Δ =α_0 ^2 −((12)/α_0 ) =((α_0 ^3 −12)/α_0 ) =((−α_o +3−12)/α_0 ) =−((α_0 +9)/α_0 ) ⇒z_1 =−α_0 +i(√((α_0 +9)/α_0 )) z_2 =−α_0 −i(√((α_0 +9)/α_0 )) and f(x) =(x−α_0 )(x−z_1 )(x−z_2 ) =(x−α_0 )(x+α_0 −i(√((α_0 +9)/α_0 )))(x+α_0 +i(√((α_0 +9)/α_0 )))
1)fcontinueandf′(x)=3x2+1>0⇒fisincreazingwehavef(1)=−1andf(2)=8+2−3=7⇒f(1).f(2)<0⇒∃α0∈]1,2[unique/f(α0)=02)f(x)=(x−α0)(x2+ax+b)⇒x3+ax2+bx−α0x2−αoax−bα0=x3+x−3⇒(a−α0)x2+(b−αoa)x−bα0=x−3⇒{a−α0=0b−α0a=1and−bα0=−3⇒a=α0andb=3α0f(x)=(x−α0)(x2+α0x+3α0)thisistbefactoruzationatR[x]x2+α0x+3α0=0→Δ=α02−12α0=α03−12α0=−αo+3−12α0=−α0+9α0⇒z1=−α0+iα0+9α0z2=−α0−iα0+9α0andf(x)=(x−α0)(x−z1)(x−z2)=(x−α0)(x+α0−iα0+9α0)(x+α0+iα0+9α0)

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