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Question

If α,β,γ,δ are the roots of the equation x4+qx2+rx+s=0 find the equation whose roots are β+γ+δ+(βγδ)1, and c.

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Solution

Given equation, x4+qx2+rx+s=0 which has the roots α,β,γ,δ

Therefore, α=0,αβ=q,αβγ=r,αβγδ=s


Now, β+γ+δ+1βγδ=(α+β+γ+δ)+ααβγδα=(1ss)α

Therefore, roots of the required equation are (1ss)α,(1ss)β,(1ss)γ,(1ss)δ,c

Or Roots of the required equation are λα,λβ,λγ,λδ,c where λ=1ss


For the required equation,

S1=λα+λβ+λγ+λδ+c=λ(α+β+γ+δ+c)=c

S2=λ2αβ+λ(α)c=λ2q

S3=λ3αβγ+λ2(αβ)c=λ3r+λ2rc

S4=λ4αβγδ+λ3(αβγ)c=λ4sλ3rc

S5=λ4αβγδc=λ4sc


Required equation is x5S1x4+S2x3S3x2+S4xS5=0

Or x5cx4+λ2qx3(λ3r+λ2rc)x2+(λ4sλ3rc)xλ4sc=0

Or s3x5s3cx4+(1s)2sqx3+(1s)2(1ssc)rx2+(1s)3(1src)x(1s)3c=0


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