\(\left(\frac{1}{2} x-y\right)^{2}=?\) A. \(\frac{1}{4} x^{2}+y^{2}\) B. \(\frac{1}{4} x^{2}-x y+y^{2}\) C. \(\frac{1}{2} x^{2}-x y+y^{2}\) D. \(x^{2}+y^{2}\) E. \(x^{2}-x y+y^{2}\)

Understanding the binomial square expansion is a fundamental aspect of algebra that can simplify many types of problems, including those found on standardized tests like the ACT. The binomial square describes the result of multiplying a binomial by itself. A binomial, in this context, is an algebraic expression with two terms. The general form of a binomial square is \( (a + b)^2 = a^2 + 2ab + b^2 \).

The pattern is straightforward: the first term and the second term are each squared, and then the product of the two terms is doubled. In the problem \( (\frac{1}{2} x - y)^2 \), \((\frac{1}{2})x\) is \(a\), and \(y\) is \(b\). By applying the binomial square pattern, we get \( (\frac{1}{2} x)^2 - 2(\frac{1}{2} x)(y) + y^2 \), which ultimately simplifies as shown in the solutions. This fundamental concept not only helps in solving quadratic equations but also lays the foundation for understanding the expansion of larger polynomials.

An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. For example, \( \frac{1}{4}x^2 - xy + y^2 \) is an algebraic expression. In the context of the ACT Math or similar mathematical problems, becoming adept at working with algebraic expressions is crucial.

These expressions can be used to represent real-world situations and solve problems where the exact values of variables are unknown. To confidently tackle algebraic expressions, you should understand the following:

• Variables are symbols, normally letters, that stand for unknown values.
• Constants are fixed numbers.
• Coefficients are numbers that multiply the variables.

Thus, recognizing the various components of algebraic expressions is the first step to manipulating and simplifying them, which brings us to our next core concept.

The process of simplifying algebraic expressions is reducing them to their simplest form while maintaining their original value. This could involve combining like terms, factoring, expanding products, or canceling common factors in fractions. The goal is to make the expression easier to understand or work with.

In the given problem, the expression \( (\frac{1}{2} x - y)^2 \) was simplified step by step. The square of \( \frac{1}{2}x \) was calculated, and the double product of \( \frac{1}{2}x \) and \( y \) was simplified. Finally, the expression was written without parentheses and arranged in a standard form. Simplifying expressions is not just about making them look neater; it is essential for solving equations and inequalities, analyzing functions, and understanding calculus later on.