06-23-2017, 11:53 AM
(This post was last modified: 06-23-2017, 11:54 AM by Dane Regan.)
Vote number (α³ + β³ + γ³) where...
The equation z³ - 5z² + 4z + 14 = 0 has the roots α, β, and γ.[spoiler]23
(α + β + γ)³ = α³ + 3α²β + 3α²γ + β³ + 3β²α + 3β²γ + γ³ + 3γ²α + 3γ²β + 6αβγ = α³ + β³ + γ³ + 3(α + β + γ)(αβ + βγ + γα) - 3αβγ
α³ + β³ + γ³ = (α + β + γ)³ - 3(α + β + γ)(αβ + βγ + γα) + 3αβγ
As z³ - 5z² + 4z + 14 = 0 = (z - α)(z - β)(z - γ)
By comparing coefficients of different powers of z:
α + β + γ = 5
αβ + βγ + γα = 4
αβγ = -14
So α³ + β³ + γ³ = 5³ - 3*5*4 + 3*(-14) = 23[/spoiler]
The equation z³ - 5z² + 4z + 14 = 0 has the roots α, β, and γ.[spoiler]23
(α + β + γ)³ = α³ + 3α²β + 3α²γ + β³ + 3β²α + 3β²γ + γ³ + 3γ²α + 3γ²β + 6αβγ = α³ + β³ + γ³ + 3(α + β + γ)(αβ + βγ + γα) - 3αβγ
α³ + β³ + γ³ = (α + β + γ)³ - 3(α + β + γ)(αβ + βγ + γα) + 3αβγ
As z³ - 5z² + 4z + 14 = 0 = (z - α)(z - β)(z - γ)
By comparing coefficients of different powers of z:
α + β + γ = 5
αβ + βγ + γα = 4
αβγ = -14
So α³ + β³ + γ³ = 5³ - 3*5*4 + 3*(-14) = 23[/spoiler]