Problem 29 - Elementary Algebra

Question

Factor completely. $$ 4 x 2 y 2-1 $$

Step-by-Step Solution

Answer

The expression factors as \((2xy - 1)(2xy + 1)\).

Step 1: Recognize the expression as a difference of squares

The given expression is \(4x^2y^2 - 1\). Notice that both terms are perfect squares. The first term, \(4x^2y^2\), can be written as \((2xy)^2\) and the second term, 1, can be written as \(1^2\). This is a difference of squares which follows the form \(a^2 - b^2\).

Step 2: Apply the difference of squares formula

The difference of squares formula is given by \(a^2 - b^2 = (a - b)(a + b)\). In this expression, take \(a = 2xy\) and \(b = 1\). Substitute into the formula: \((2xy - 1)(2xy + 1)\).

Step 3: Verify the factorization

Multiply the factors back to check correctness: \((2xy - 1)(2xy + 1) = (2xy)^2 + (2xy)(1) - (2xy)(1) - 1^2 = 4x^2y^2 - 1\). This confirms the factorization is accurate.

Key Concepts

Factorization

Factorization is a powerful tool in algebra, used to write expressions as a product of simpler terms. When you encounter an expression like a difference of squares, such as the one in the problem, factorization helps to break it down. The key idea is to express a complex polynomial as a product of simpler polynomials, making it easy to solve or simplify algebraic equations.

Here's how factorization works with the difference of squares:

• Identify terms that are perfect squares.
• Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).

In our example, we have \(4x^2y^2 - 1\). Both \(4x^2y^2\) as \( (2xy)^2 \) and 1 as \(1^2\) are perfect squares. Thus, factorization involves recognizing the difference of squares pattern and breaking it into \((2xy - 1)(2xy + 1)\). This process simplifies the expression by transforming the original expression into a product, unveiling its structure and making solving easier.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and mathematical operations like addition, multiplication, and subtraction. They form the foundation for algebra, allowing us to express mathematical relationships and solve problems effectively. In the expression \(4x^2y^2 - 1\), we deal with variables \(x\) and \(y\), coefficients like 4, and an operation illustrating subtraction.

Understanding algebraic expressions involves recognizing patterns and working with them, helping us to:

• Define the relationships between different quantities.
• Manipulate and simplify expressions for easier problem-solving.
• Develop an understanding of how various algebraic rules, such as the difference of squares, apply.

Algebraic expressions are integral in math, allowing us to transition from arithmetic and explore more complex algebraic concepts..

Perfect Squares

Perfect squares are numbers or algebraic expressions that are the square of an integer or another expression. For instance, the expression \(4x^2y^2\) is a perfect square because it equals \((2xy)^2\). Similarly, 1 is a perfect square because it equals \(1^2\).

Identifying perfect squares is crucial because it allows us to use formulas like the difference of squares to simplify expressions effectively.

• Each perfect square has an integer or simpler expression that, when multiplied by itself, equals the original term.
• Recognizing perfect squares in algebraic expressions makes factorization and solving equations more straightforward.

In our exercise, spotting \(4x^2y^2\) and 1 as perfect squares enabled the straightforward application of the difference of squares formula, simplifying the given expression considerably.