Roots of a cubic polynomial

Looking for a quick identity to deal with the sum of the cubes of the roots of a cubic.

I know that a² + b² + c² = (a + b + c)² - 2(ab + ac + bc)

and i am trying to figure out one for

a³ + b³ + c³

my teacher has it as a³ + b³ + c³ = (a + b + c)³ - 3(ab + ac + bc)

but when i tried expanding the RHS i got

a³ + b³ + c³ + 3a²b + 3a²c +3ab² + 3ac² + 3b²c + 3bc² - 3ab - 3ac - 3bc + 6abc

which isnt even close to the identity we are after

I have tried to come up it but haven't had any luck yet.

thanks for any help