Question

If a + b + c = 14, and ab + bc + ca = 65, then find the value of a³ + b³ + c³ - 3abc.
1. -49
2. 49
3. 14
4. -14

Answer 1

Correct Answer - Option 3 : 14

Given:

a + b + c = 14

ab + bc + ca = 65

Identity used:

a3 + b3 + c3 - 3abc = (a + b + c) × [(a + b + c)² - 3 × (ab + bc + ca)]

Calculation:

a3 + b3 + c3 - 3abc = (a + b + c) × [(a + b + c)2 - 3 × (ab + bc + ca)]

⇒ a3 + b3 + c3 - 3abc = 14 × [(14)² - 3 × 65]

⇒ a3 + b3 + c3 - 3abc = 14 × [196 - 195]

⇒ a3 + b3 + c3 - 3abc = 14 × 1

∴ a3 + b3 + c3 - 3abc is 14