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Question

# If the polynomial P(x)=24x4+λ1x3+λ2x2+λ3x+1, where λ1,λ2,λ3∈R has four positive real roots α,β,γ,δ such that α+2β+3γ+4δ=4, then

A
λ1+λ2+λ3=25
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B
λ1=50
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C
λ3=10
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D
λ3=25
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Solution

## The correct options are A λ1+λ2+λ3=−25 B λ1=−50 C λ3=−10α,β,γ,δ are roots of the polynomial P(x)=24x4+λ1x3+λ2x2+λ3x+1 ∴αβγδ=124 and α+2β+3γ+4δ=4 We know, A.M≥G.M.⇒α+2β+3γ+4δ4≥(α⋅2β⋅3γ⋅4δ)1/4 But, α+2β+3γ+4δ4=(α⋅2β⋅3γ⋅4δ)1/4 Therfore, α=2β=3γ=4δ=1 ⇒α=1, β=12, γ=13,δ=14 α+β+γ+δ=−λ124⇒λ1=−50 αβγ+βγδ+γδα+δαβ=−λ324⇒αβγδ(1α+1β+1γ+1δ)=−λ324⇒124(1+2+3+4)=−λ324⇒λ3=−10 ∵α is a root of the given polynomial. ∴24+λ1+λ2+λ3+1=0⇒λ1+λ2+λ3=−25 ∴λ2=35

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