%5E2-expand.jpeg "(x-y)^2 Expand")
Understanding the Expansion of (x-y)²
The expression (x-y)² represents the square of the binomial (x-y). Expanding this expression is a fundamental concept in algebra and is essential for various mathematical operations.
Understanding the Concept
The square of a binomial means multiplying the binomial by itself. Therefore:
(x - y)² = (x - y) * (x - y)
To expand this, we need to distribute each term of the first binomial to each term of the second binomial.
The Expansion Process
- Multiply the first terms: x * x = x²
- Multiply the outer terms: x * -y = -xy
- Multiply the inner terms: -y * x = -xy
- Multiply the last terms: -y * -y = y²
Now, we have: x² - xy - xy + y²
Simplifying the Expression
Combining like terms, we get the final expanded form:
(x - y)² = x² - 2xy + y²
Key Points to Remember
- The sign of the middle term is always negative because the product of a positive and a negative term is negative.
- The coefficient of the middle term is always twice the product of the two terms in the binomial.
- The expansion of (x-y)² is a perfect square trinomial, meaning it can be factored back into the original binomial.
Applications of the Expansion
The expansion of (x-y)² is used extensively in:
- Factoring polynomials: Knowing this expansion helps recognize and factor perfect square trinomials.
- Solving equations: The expansion can be used to simplify equations involving squared binomials.
- Calculus: Derivatives and integrals often involve expanding binomials.
- Geometry: The expansion is used to derive formulas for areas and volumes of geometric shapes.
Conclusion
Understanding the expansion of (x-y)² is essential for a strong foundation in algebra. It allows for efficient manipulation of expressions, simplifying equations, and applying mathematical concepts in various fields.