Question

If a + b + c =9 and ab + bc +ca =23, then what is a^3 + b^3 + c^3

Answer 1

a + b + c = 9 ...(i)

(a + b + c)² = 81

ab + bc + ca = 23 ... (ii)

a² + b² + c² + 2(ab + bc + ca) = 81

a² + b² + c² + 2(23) = 81

a² + b² + c² + 46 = 81

a² + b² + c² = 35 ... (iii)

a³ + b³ + c³ - 3abc

= (a + b + c)[a² + b² + c² - (ab + bc + ca)]

= (9)(35 - 23) ... (from i, ii and iii)

= (9)(12)

= 108

Hence, if a + b + c = 9, ab + bc + ca = 23, then a³ + b³ + c³ - 3abc = 108.