If $a+b+c=0,$  then $a^3+b^3+c^3$  is equal to

Question: If a + b + c = 0, then what is the value of a³ + b³ + c³?

Explanation

We start with the identity:

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

Here, the term (a + b + c) appears in the factorization. If a + b + c = 0, then the entire right-hand side of the equation becomes zero because it is multiplied by (a + b + c).

This implies:

a³ + b³ + c³ - 3abc = 0

Rearranging the terms, we get:

a³ + b³ + c³ = 3abc

Conclusion

Thus, when a + b + c = 0, the value of a³ + b³ + c³ simplifies to 3abc.

Key Points to Remember:

• The identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac) is crucial for solving such problems.
• When a + b + c = 0, the left-hand side directly simplifies to 3abc.