Question

alpha+beta+gamma=2,alpha^2+beta^2+gamma^2=6, alphq^3+beta^3+ gamma^3 = 8 then alpha^4+beta^4+gamma^4 =?

Answer 1

α + β + γ = 2
α² + β² + γ² =6
α³ + β³ + γ³ = 8

then,
( α + β+ γ)² = α²+ β²+ γ² +2( αβ+ βγ + γα)
(2)² = 6 +2(αβ+ βγ + γα)
αβ + βγ + γα =-1

we also know,
α³ + β³ + γ³ -3 αβγ = ( α + β+ γ)( α²+ β²+ γ²- αβ- βγ- γα)

8 -3 αβγ = 2× ( 6 +1 )

αβγ = -2


now,
( α²+ β²+ γ²)² = α^4 + β^4 + γ^4 + 2( α² β²+ β²γ²+ γ² α²)

(6)² = α^4+ β^4+ γ^4 +2{ ( αβ+ βγ+ γα)²-2αβγ( α + β + γ)}

36 = α^4+ β^4+ γ^4 +2{(-1)² -2(-2)(2)}

36 - 18 = α^4+ β^4+ γ^4

α^4+ β^4+ γ^4 = 18

Answer 2

I have solved it by a generalized method...SO, try to equipped it with.Obviously, in these types of questions you wouldn't like to invest more time.