Given, α,β,γ are the real roots of x3+ax2+bx+c=0, where a,b,c∈R and a,b≠0 ∴α+β+γ= Sum of roots taken one at a time =−a αβ+βγ+γα= Sum of roots taken two at a time =b αβγ= Product of roots =−c Also, given system of equations in u,v,w
αu+βv+γw=0
βu+γv+αw=0
γu+αv+βw=0
} has non-trivial solution. ∴∆=0⇒|
α
β
γ
β
γ
α
γ
α
β
|=0 ⇒α(βγ−α2)−β(β2−γα)+γ(αβ−γ2)=0 (expanding along R1 ) ⇒αβγ−α3−β3+αβγ+αβγ−γ3=0 ⇒α3+β3+γ3=3αβγ Then (using standard result), Either α+β+γ=0 or α=β=γ If α+β+γ=0, then −a=0 ⇒a=0 which is not possible according to given condition. ∴α+β+γ=0 (not possible) Now, α+β+γ=−a ⇒α+α+α=−a(∵α=β=γ) ⇒a=−3α...(i) αβ+βγ+γα=b ⇒b=3α2...(ii) Using Eqs. (i) and (ii),