Deriving the Formula for a³ + b³ + c³ - 3abc
The formula for a³ + b³ + c³ - 3abc is a useful algebraic identity, particularly in factoring and simplifying expressions. It is derived using the following steps:
Step 1: Factoring the Sum of Cubes
Recall the formula for the sum of cubes:
a³ + b³ = (a + b)(a² - ab + b²)
Step 2: Expressing a³ + b³ + c³ - 3abc
We can rewrite the expression a³ + b³ + c³ - 3abc as:
(a³ + b³) + c³ - 3abc
Step 3: Applying the Sum of Cubes Formula
Using the formula from Step 1, we can factor (a³ + b³) + c³ - 3abc as:
(a + b)(a² - ab + b²) + c³ - 3abc
Step 4: Rearranging Terms
Rearrange the terms within the expression:
(a + b)(a² - ab + b²) + c³ - 3abc = (a + b)(a² - ab + b²) + (c³ - 3abc)
Step 5: Factoring c³ - 3abc
We can factor c³ - 3abc by taking out a common factor of c:
c³ - 3abc = c(c² - 3ab)
Step 6: Combining the Expressions
Substituting the result from Step 5, we get:
(a + b)(a² - ab + b²) + c(c² - 3ab)
Step 7: Factoring the Expression
Finally, we can factor out a common factor of (a² - ab + b² - 3ab), leading to the final formula:
(a + b)(a² - ab + b²) + c(c² - 3ab) = (a + b + c)(a² + b² + c² - ab - ac - bc)
Therefore, we have derived the formula:
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ac - bc)
Applications of the Formula
This formula is frequently used in:
- Factoring algebraic expressions
- Solving equations
- Simplifying complex expressions
The formula helps in simplifying expressions and finding solutions more efficiently.
