(a%5E2%2Bab%2Bb%5E2)-formula-name.jpeg "(a-b)(a^2+ab+b^2) Formula Name")
The Formula: (a - b)(a² + ab + b²) = a³ - b³
This formula is known as the difference of cubes formula. It is a fundamental algebraic identity that helps simplify expressions and solve equations involving the difference of two cubes.
Understanding the Formula
The formula states that the product of the difference of two terms (a - b) and the sum of the squares of the first term and the product of the two terms, plus the square of the second term (a² + ab + b²), is equal to the difference of the cubes of the two terms (a³ - b³).
Applications
The difference of cubes formula is widely used in algebra, trigonometry, and calculus. Some common applications include:
- Factoring expressions: It allows you to factor expressions that contain the difference of two cubes into simpler expressions.
- Solving equations: It can be used to solve equations involving the difference of two cubes.
- Simplifying expressions: It helps simplify complex expressions by reducing them to a simpler form.
Example
Let's consider an example to illustrate the use of the difference of cubes formula.
Suppose we want to factor the expression (x³ - 8). We can recognize that this expression is in the form of (a³ - b³), where a = x and b = 2. Applying the difference of cubes formula, we get:
(x³ - 8) = (x - 2)(x² + 2x + 4)
Therefore, we have factored the expression (x³ - 8) into two simpler factors.
Conclusion
The difference of cubes formula is a powerful tool in algebra that simplifies expressions and solves equations involving the difference of two cubes. Its applications are numerous, making it an essential formula to remember.