algebric identity

Show that if if a+ b + c = 5 and ab + bc + ca = 10,

 then a³ + b³ + c³ -3abc = -25

Answer:

a+ b + c = 5

or, a+ b = 5 – c 

taking cube on both sides

a³ + b³ + 3ab(b+a) =125-c³ +15c² – 75c

or, a³ + b³ + c³ +3ab(5 – c) = 125 +15c² -75c

or, a³ + b³ + c³ -3abc =125 +15c² -75c -15ab

or, a³ + b³ + c³ -3abc =125 +15c(5 – a – b) -75c -15ab

or, a³ + b³ + c³ -3abc =125 +75c -15ac – 15bc  - 75c -15ab

or, a³ + b³ + c³ -3abc = 125 -15( ab + bc + ca)

or, a³ + b³ + c³ -3abc = 125 – 150 = - 25