If x^3 +ax^2 +bx+c = 0 has the roots are α^� β and γ . find the value of αβ^2 +βγ^2 +γα^2 in terms a,b and c


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Question Number 102883 by bobhans last updated on 11/Jul/20

$I f x^{3} + {a x}^{2} + b x + c = 0 h a s t h e r o o t s a r e$ $\bar{α} β a n d γ . f i n d t h e v a l u e o f$ $α β^{2} + β γ^{2} + γ α^{2} i n t e r m s a, b a n d c$
	Answered by bemath last updated on 11/Jul/20

	$α + β + γ = - a$ $α β + α γ + β γ = b$ $α β γ = - c$ $\Rightarrow (α + β + γ) (α β + α γ + β γ) = - a b$ $\Rightarrow α^{2} β + + α^{2} γ + α β γ + α β^{2} + α β γ + β^{2} γ + α β γ + α γ^{2} + β γ^{2} = - a b$ $α β^{2} + β γ^{2} + γ α^{2} + α^{2} β$ $+ β^{2} γ + α γ^{2} + 2 α β γ = - a b$ $α β^{2} + β γ^{2} + γ α^{2} =$ $- a b - 2 (- c) - β^{2} γ - α γ^{2} - α^{2} β$ $= - a b + 2 c - (β^{2} γ + α γ^{2} + α^{2} β)$
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