Question

if a+b+c=2,ab+bc+ca=-1 and abc=-2 find the value ofa^3+b^3+c^3

Answer 1

Firstly,
(a + b + c)^2 = a^2+b^2+c^2 +2 (ab + bc + ca)
=> 2^2 = a^2+b^2+c^2 + 2×(-1)
=> a^2+b^2+c^2 = 4-(-2) = 6

Then we use the idendity that
a^3+b^3+c^3 - 3abc =
(a+b+c)(a^2+b^2+c^2 - ab - bc - ac).

So, a^3+b^3+c^3 = (a+b+c)(a^2+b^2+c^2 - ab - bc - ac) + 3abc

Putting the values,
a^3+b^3+c^3 = 2 (6-(-1)) + 3×2
a^2+b^2+c^2 = 14 + 6 = 20
Hence the ans. is 20