## 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.