If a + b + c = 1, ab + bc = ca = -1 and abc = -1, then what is the value of $a^3 + b^3 + c^3 $?

1

If a + b + c = 1,

ab + bc = ca = -1

abc = -1

Then what is the value of $a^3 + b^3 + c^3 $?

We know that,

(a + b + c)² = a² + b² + c² + 2(ab + bc + ac)

(1)² = a² + b² + c² + 2(-1)

 a² + b² + c² = 3

We also know that,

a³ + b³ + c³ - 3abc = ( a + b + c  ) ( a² + b² + c² - (ab + bc + ca))

a³ + b³ + c³ - 3(-1) = ( 1  ) ( 3- (-1))

a³ + b³ + c³ = 4 - 3 = 1