(x%5E2%2Bxy%2By%5E2)-formula.jpeg "(x-y)(x^2+xy+y^2) Formula")
Understanding the (x-y)(x^2 + xy + y^2) Formula
The formula (x-y)(x^2 + xy + y^2) is a special case of a more general algebraic concept known as the difference of cubes. It provides a quick and efficient way to factor expressions that fit a specific pattern.
The Difference of Cubes Formula
The general formula for the difference of cubes is:
a³ - b³ = (a - b)(a² + ab + b²)
This formula states that the difference of two cubes can be factored into the product of the difference of their cube roots and a quadratic expression.
Applying the Formula to (x-y)(x^2 + xy + y^2)
In our specific case, we can see that:
- a = x
- b = y
Therefore, applying the difference of cubes formula, we get:
(x³ - y³) = (x - y)(x² + xy + y²)
How to Use the Formula
The formula is useful for:
- Factoring expressions: If you encounter an expression in the form a³ - b³, you can immediately factor it using the formula.
- Simplifying expressions: By factoring, you can often simplify complex expressions, making them easier to work with.
- Solving equations: The formula can be used to solve equations involving the difference of cubes.
Example
Factor the expression x³ - 8
- Recognize that 8 is the cube of 2 (2³ = 8).
- Apply the difference of cubes formula:
- a = x
- b = 2
- (x³ - 2³) = (x - 2)(x² + 2x + 4)
Therefore, the factored form of x³ - 8 is (x - 2)(x² + 2x + 4).
Conclusion
The (x-y)(x² + xy + y²) formula, derived from the difference of cubes formula, provides a valuable tool for factoring and simplifying expressions. Understanding this formula can save you time and effort when working with algebraic expressions.