Question

If `alpha`, `beta`, `gamma` are such that `alpha+beta+gamma=2`, `alpha^2+beta^2+gamma^2=6`, `alpha^3+beta^3+gamma^3=8`, then `alpha^4+beta^4+gamma^4`

Answer 1

We have,
`(alpha+beta+gamma)^(2)=alpha^(2)+beta^(2)+gamma^(2)+2(betagamma+gammaalpha+alphabeta)`
`rArr 4 = 6 + 2 (betagamma+gammaalpha+alphabeta)`
`rArr betagamma+gammaalpha+alphabeta = -1`
Also, `alpha^(3) + beta^(3) + gamma^(3) - 3alphabeta gamma`
`=(alpha+beta+gamma)(alpha^(2)+beta^(2)+gamma^(2))(betagamma+gammaalpha+alphabeta)`
or " " 8 - 3 `alpha+beta+gamma = 2(6+1)`
or `3alpha+beta+gamma = 8 - 14 = -6`
or `alpha+beta+gamma = - 2`
Now, `(alpha^(2)+beta^(2)+gamma^(2))^(2)=Sigmaalpha^(4)+2Sigmaalpha^(2)beta^(2)`
`=Sigmaalpha^(4)+2[(Sigmaalphabeta)^(2)-2alphabetagamma(Sigmaalpha)]`
or `=Sigmaalpha^(4)= 36 - 2 [(-1)^(2)-2(-2)(2)] = 18`