Factorize: a³ - b³ - c³ - 3abc
Given Expression
a³ - b³ - c³ - 3abc
Solution
We know the identity:
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
The given expression is:
a³ - b³ - c³ - 3abc
Rewrite it as:
a³ + (-b)³ + (-c)³ - 3a(-b)(-c)
Comparing with the identity, let:
Substitute these values into the identity:
(a - b - c)
[a² + (-b)² + (-c)² - a(-b) - (-b)(-c) - (-c)a]
Simplifying:
(a - b - c)(a² + b² + c² + ab - bc + ac)
Final Answer
a³ - b³ - c³ - 3abc =
(a - b - c)(a² + b² + c² + ab + ac - bc)