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)))

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 )))