If - - - are the roots of the equation x3-ax2-bx-c-0 then 1- 2 1- 21- 2-

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Discriminant

If α, β, ...

Question

If $α$ , $β$ , $γ$ are the roots of the equation $x^{3} + a x^{2} + b x + c = 0$ then $(1 + α^{2}) (1 + β^{2}) (1 + γ^{2}) =$

(c - a)^{2} + (b - 1)^{2}

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(c + a)^{2} + (b + 1)^{2}

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(c - a)^{2} - (b - 1)^{2}

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(c + a)^{2} - (b + 1)^{2}

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Solution

The correct option is C $(c - a)^{2} + (b - 1)^{2}$
$x^{2} + a x^{2} + b x + c = 0$
Roots are $α, β, γ$
$α + β + γ = - a$
$α β + β γ + γ α = b$
$α β γ = - c$
$1 + α^{2} + β^{2} + γ^{2} + α^{2} γ^{2} + β^{2} γ^{2} + α^{2} β^{2} + α^{2} β^{2} γ^{2}$
$= 1 + {(α + β + γ)}^{2} - 2 (α β + β γ + γ α) + {(α β + β γ + γ α)}^{2} - 2 α β γ (α + β + γ) + {(α β γ)}^{2}$
$1 + a^{2} - 2 b + b^{2} - 2 a c + c^{2}$
${(c - a)}^{2} + {(b - 1)}^{2}$

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