If-x-3-ax-2-bx-c-0-has-the-roots-are-and-find-the-value-of-2-2-2-in-terms-a-b-and-c-

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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