(a%5E2%2Bab%2Bb%5E2)-formula.jpeg "(a-b)(a^2+ab+b^2) Formula")
The Difference of Cubes Formula: (a - b)(a² + ab + b²)
The formula (a - b)(a² + ab + b²) = a³ - b³ is known as the difference of cubes formula. It provides a quick and efficient way to factorize expressions of the form a³ - b³.
Understanding the Formula
The formula is derived from the expansion of the product:
(a - b)(a² + ab + b²) = a(a² + ab + b²) - b(a² + ab + b²)
Expanding this gives:
a³ + a²b + ab² - a²b - ab² - b³
Simplifying, we get:
a³ - b³
Using the Formula
The difference of cubes formula can be applied to factorize expressions where both terms are perfect cubes.
Here's how to use it:
- Identify the terms: Determine the cube root of each term in the expression.
- Apply the formula: Substitute the cube roots (a and b) into the formula.
- Simplify: The result will be the factored form of the expression.
Example:
Factorize the expression: x³ - 8
- Identify the terms: The cube root of x³ is x, and the cube root of 8 is 2.
- Apply the formula: Substitute a = x and b = 2 into the formula: (x - 2)(x² + 2x + 2²)
- Simplify: The factored form is (x - 2)(x² + 2x + 4)
Applications
The difference of cubes formula has various applications in algebra and other mathematical fields, including:
- Simplifying expressions: It helps simplify complex expressions involving cubes.
- Solving equations: The formula can be used to solve equations with cubic terms.
- Calculus: The formula aids in differentiating and integrating expressions with cubic terms.
Conclusion
The difference of cubes formula is a valuable tool for factoring expressions and simplifying calculations. Mastering this formula allows you to efficiently manipulate algebraic expressions and solve problems involving cubes.