An algebra problem by A Former Brilliant Member
If α , β and γ are roots of equation x 3 + 4 x + 1 .Then, find ( α + β 1 + γ 2 α + β ) + ( β + γ 1 + α 2 β + γ ) + ( γ + α 1 + β 2 γ + α )
The answer is 8.
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2 solutions
Nice solution! +1 .....Firstly you got it luckily :p ....Typo: α β + β γ + γ α = 4 and α β γ = − 1 :)
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Lol I first time read it as ( α + β 1 − γ 2 α + β ) and that's why I answered 0 and then saw that the problem was deleted :-)..... Anyways typo in this problem was commited since I was writing a solution to another problem :-)
Hi Abhay Kumar. I am just letting you know you forgot to write " = 0 " in your problem.
Did the same !
Let α + β 1 + γ 2 α + β = γ − 2 = y (as you stated). Then y − 2 = γ and since that is a root of the given function: ( y − 2 ) 3 − y 8 + 1 = 0 multiply by y 3 and simplify: y 3 − 8 y 2 − 8 = 0 . The sum of the roots of this function is 8, which is the desired result.
Same way bud
Relevant wiki: Vieta's Formula Problem Solving - Intermediate
From Vieta's formulas we get: α + β + γ = 0 ⎩ ⎪ ⎨ ⎪ ⎧ α + β = − γ α + γ = − β β + γ = − α ⟹ cyc ∑ ( α + β 1 + γ 2 α + β ) = cyc ∑ γ − 2 = − 2 ( α 1 + β 1 + γ 1 ) x = x 1 ⟹ ( x 1 ) 3 + 4 ( x 1 ) + 1 = 0 ⟹ x 3 + 4 x 2 + 1 = 0 ⟹ α 1 + β 1 + γ 1 = − 4 ⟹ − 2 ( α 1 + β 1 + γ 1 ) = 8
Nice solution (+1) :-)
You can also get the sum of the reciprocals by dividing ab + bc + ca by abc.
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Indeed :),but I wanted to do something different and this method may seem slightly quicker to some people.
See Vieta's formula α β + β γ + γ α = 4 , α β γ = − 1 α + β + γ = 0 ⟹ α + β = − γ
⎝ ⎜ ⎜ ⎛ − γ α + β 1 + γ 2 α + β − γ ⎠ ⎟ ⎟ ⎞ = γ − 2 The required expression is:-
α − 2 + β − 2 + γ − 2 = = − 2 ( α β γ α β + β γ + γ α ) − 2 × ( − 1 ) 4 = 8