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Question

If α,β,γ are the roots of the equation x33x+11=0, what is the equation whose roots are (α+β),(β+γ) and (γ+α)?

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Solution

given equation x33x+11=0
α+β+γ=0 α,β,γ are roots
αβ+βγ+γα=3 of the equation
αβγ=11
if α+β,β+γ,γ+α are roots then
Equation say ax3+bx2+cx+d=0
(α+β)+(β+γ)+(γ+α)=b/a.
2(α+β+γ)=b/ab/a=0
(α+β)(β+γ)+(β+γ)(γ+α)+(γ+α)(α+β)=c/a
We know that α+β+γ=0
α+β=γ β+γ=α γ+α=β
(γ×α)+(β×α)+(β×γ)=c/a
αβ+βγ+γα=c/a
c/a=3
(α+β)(β+γ)(γ+α)=d/a
αβγ=d/a
d/a=11
ax3+bx2+cx+d=0x3+bax2+cax2+cax+da=0
Req. Equation is x33x11=0

1170114_528967_ans_2d8bca7ac0ca4cc785e8cb5e2b021a02.jpg

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