The a³ + b³ Formula
The formula a³ + b³ is a fundamental algebraic identity that is used to factor the sum of two cubes. It states that:
a³ + b³ = (a + b)(a² - ab + b²)
This formula can be derived using the following steps:
- Start with the factorization of a³ - b³: We know that (a - b)(a² + ab + b²) = a³ - b³.
- Substitute -b with b: If we replace -b with b in the above equation, we get (a + b)(a² - ab + b²) = a³ + b³.
This formula is often used to factor expressions that contain the sum of two cubes. For example, we can use this formula to factor the expression x³ + 8.
Here's how:
- Identify the cubes: x³ and 8 are both cubes, where x is the cube root of x³ and 2 is the cube root of 8.
- Apply the formula: Substitute a = x and b = 2 into the formula:
- x³ + 8 = (x + 2)(x² - 2x + 4)
Therefore, the factored form of x³ + 8 is (x + 2)(x² - 2x + 4).
Applications of the a³ + b³ Formula:
The a³ + b³ formula has various applications in algebra and calculus, including:
- Simplifying expressions: The formula can simplify complex expressions involving the sum of two cubes.
- Solving equations: The formula can be used to solve equations that involve the sum of two cubes.
- Calculus: The formula can be used to differentiate and integrate functions that involve the sum of two cubes.
Example:
Simplify the following expression using the a³ + b³ formula:
x³ + 27
- Identify the cubes: x³ and 27 are both cubes, where x is the cube root of x³ and 3 is the cube root of 27.
- Apply the formula: Substitute a = x and b = 3 into the formula:
- x³ + 27 = (x + 3)(x² - 3x + 9)
Therefore, the simplified form of x³ + 27 is (x + 3)(x² - 3x + 9).
By understanding and applying the a³ + b³ formula, you can solve various algebraic problems and simplify complex expressions efficiently.
