Question

If α , β, γ are the roots of the equation
x 3 − 6 x 2 + 11 x + 6 = 0, then
∑ α 2β + ∑ αβ 2 is equal to

Answer 1

x³-6x²+11x+6=0
let a, b, c be the roots of the equation.
s1=a+b+c=6
s2=ab+bc+ca=11
s3=abc=-6
a²b+a²c+bc²+ab²+ac²+b²c=?
from s3,
ab=-6/c or ac=-6/b or bc=-6/a
putting above values one by one in s2, we get
a²b+a²c-6=11a...............eq 1
ac²+bc²-6=11c................eq2
ab²+b²c-6=11b...............eq3
adding all 3 eq, we get
a²b+a²c+bc²+ab²+ac²+b²c-18=11a+11b+11c
=11(a+b+c)=11s1=11×6=66
therefore, the answer will be 66+18=84

note: here i replaced α,β with a, b

hope it helps you.