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Question

If α,β are roots of the equation ax2+bx+c=0, where a,b,c are distinct real values, then (1+α+α2)(1+β+β2) is

A
1[(b+c2ca)(b2+c2+bca2)]
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B
1[(b+c2ca)(b2+c2+bca3)]
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C
1[(a+c2ca)(a2+c2+bca2)]
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D
1[(a+b2ca)(c2+b2+aca2)]
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Solution

The correct option is A 1[(b+c2ca)(b2+c2+bca2)]
α+β=ba

αβ=ca

(1+α+α2)(1+β+β2)

=1+β+β2+α+αβ+αβ2+α2+α2β+α2β2

=1+(α+β)+(α2+β2)+αβ+α2β2+αβ(α+β)

=1+(α+β)+[(α+β)22αβ]+αβ+α2β2+αβ(α+β)

=1ba+b2a2+2caca+c2a2ca(ba)

=1[(b+c2ca)(b2+c2+bca2)]

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