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Question

Prove that α3(αβ)(αγ)(αδ)=1δ3(δα)(δβ)(δγ).

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Solution

We have, by actual division
x3(xα)(xβ)(xγ)=1+f(x)(xα)(xβ)(xγ)
Since 11.1.1=1
where f(x) is polynomial of second degree.
Let x3(xα)(xβ)(xγ)=1+Axα+Bxβ+Cxγ ...(i)
A=limxα(xα)[x3(xα)(xβ)(xγ)] =limxα[x3(xβ)(xγ)] =α3(αβ)(αγ)
B=limxβ(xβ)[x3(xα)(xβ)(xγ)] =limxβ[x3(xα)(xγ)] =β3(βα)(βγ)
and C=limxγ(xγ)[x3(xα)(xβ)(xγ)] =limxγ[x3(xα)(xβ)] =γ3(γα)(γβ)
Substituting the values of A, B, C in (i), we have
x3(xα)(xβ)(xγ)=1+α3(αβ)(αγ)(xα)+β3(βα)(βγ)(xβ)
+γ3(γα)(γβ)(xγ)
which are required partial fractions.
x3(xα)(xβ)(xγ)=1α3(αβ)(αγ)(αx)β(βα)(βγ)(βx)
γ3(γα)(γβ)(γδ) =1α3(αβ)(αγ)(αδ)
Hence α3(αβ)(αγ)(αδ)=1δ3(δα)(δβ)(δγ)

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