Problem 54 - Algebra A Combined Function

Question

Multiply. $$ (2 x-y)(2 x+y) $$

Step-by-Step Solution

Answer

The result is \(4x^2 - y^2\).

Step 1: Identify the Pattern

First, recognize that the expression \((2x-y)(2x+y)\) is in the form \((a-b)(a+b)\), which represents the difference of squares pattern \((a^2 - b^2)\).

Step 2: Apply the Difference of Squares Formula

Using the formula \((a-b)(a+b) = a^2 - b^2\), substitute \(a = 2x\) and \(b = y\). This gives: \((2x)^2 - (y)^2\).

Step 3: Calculate and Simplify

Now calculate \((2x)^2\) and \(y^2\). \((2x)^2 = 4x^2\) and \(y^2\) remains \(y^2\). Therefore, the expression becomes \(4x^2 - y^2\).

Key Concepts

Difference of Squares Pattern

In algebra, the difference of squares is a special algebraic pattern used to simplify expressions quickly. This pattern is based on the formula \((a-b)(a+b) = a^2 - b^2\). It works because the middle terms in the expanded form cancel each other out. For example, when we multiply \((2x-y)\) and \((2x+y)\), we notice it fits this pattern:

• If \(a = 2x\), then \(a^2\) is \((2x)^2 = 4x^2\).
• If \(b = y\), then \(b^2\) is \(y^2\).

When these are plugged into the formula, the expression simplifies to \(4x^2 - y^2\). Understanding this pattern helps with quick computations.

Polynomial Multiplication

Multiplying polynomials is a key skill in algebra, often using the distributive property. Essentially, each term in one polynomial multiplies by each term in the other. For instance, with polynomials \((2x-y)\) and \((2x+y)\), you distribute as follows:

• \(2x \times 2x = 4x^2\)
• \(2x \times y = 2xy\)
• \(-y \times 2x = -2xy\)
• \(-y \times y = -y^2\)

The middle terms, \(2xy\) and \(-2xy\), cancel each other, leaving you with \(4x^2 - y^2\). This shows how polynomial multiplication ties into the alternate solution using the difference of squares.

Factoring Expressions

Factoring is the process of breaking down expressions into products of simpler expressions. This technique is invaluable when simplifying equations or solving polynomial expressions.To factor a difference of squares, you reverse the difference of squares formula:

• If you start with \(4x^2 - y^2\), you identify it as \(a^2 - b^2\), implying \((a-b)(a+b)\).
• You determine \(a = 2x\) and \(b = y\), thus factoring to \((2x-y)(2x+y)\).

Factoring is a powerful tool for simplifying and solving polynomial equations by making them more manageable.

Understanding Mathematical Expressions

Mathematical expressions are combinations of variables, numbers, and operators that represent a value. Understanding them is crucial in algebra, especially as they can be manipulated in various ways to solve problems.Expressions like \((2x-y)(2x+y)\) can be simplified or expanded depending on what you're trying to achieve:

• To simplify: Identify patterns such as difference of squares to make calculations faster.
• To expand: Use polynomial multiplication to understand the structure of the expression.

Deep knowledge of mathematical expressions allows you to navigate algebraic problems more efficiently, breaking them down into simpler components.