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Question

If α,β,γ are the roots of x3+ax2+bxx+c=0, then find α2β+αβ2.

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Solution

Given that,
α,β,γ are the roots of x3+ax2+bx+c=0

α+β+γ=a

αβ+βγ+γα=b

and αβγ=c

α2β+αβ2

=α2β+β2γ+γ2α+αβ2+βγ2+γα2

=αβ(α+β)+βγ(β+γ)+γα(γ+α)

=αβ(aγ)+βγ(aα)+γα(aβ)

=aαβαβγaβγαβγaγααβγ

=a(αβ+βγ+γα)3αβγ

=ab3(c)=ab+3c

=3cab (Ans)

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