pramrok62

pramrok62

Answered

2022-09-03

α , β , γ are roots of cubic equation x 3 + 4 x 1 = 0

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Recalculate according to your conditions!

Answer & Explanation

Samantha Braun

Samantha Braun

Expert

2022-09-04Added 9 answers

Your calculation leads us
( β + 1 ) ( γ + 1 ) α 2 + ( γ + 1 ) ( α + 1 ) β 2 + ( α + 1 ) ( β + 1 ) γ 2
= 6 { 1 ( α + 1 ) α 2 + 1 ( β + 1 ) β 2 + 1 ( γ + 1 ) γ 2 } .
Here, by Vieta's formulas, notice that
7 6 = 1 α + 1 + 1 β + 1 + 1 γ + 1 = ( α β + β γ + γ α ) + 2 ( α + β + γ ) + 3 ( α + 1 ) ( β + 1 ) ( γ + 1 ) .
Since α + β + γ = 0 , we have
( α + 1 ) ( β + 1 ) ( γ + 1 ) = 6.
Noting that your last { } equals to
( α 3 + β 3 + γ 3 ) + ( α 2 + β 2 + γ 2 ) ( α β γ ) 2 ( α + 1 ) ( β + 1 ) ( γ + 1 ) ,
we can use
α 3 + β 3 + γ 3 = 3 α β γ + ( α + β + γ ) ( α 2 + β 2 + γ 2 α β β γ γ α ) .
with
α + β + γ = 0 , α β + β γ + γ α = 4 , α β γ = 1
and
α 2 + β 2 + γ 2 = ( α + β + γ ) 2 2 ( α β + β γ + γ α ) .
Now you'll be able to get the answer.

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