Problem 12 - Technical Mathematics with Calculus

Question

Factor completely.$$4 x^{2}-9 y^{2}$$

Step-by-Step Solution

Answer

\((2x + 3y)(2x - 3y)\)

Step 1: Identify the Pattern

Recognize that the given expression is in the form of a difference of squares, \(a^2 - b^2\), where \(a = 2x\) and \(b = 3y\).

Step 2: Write in the Difference of Squares Form

Express the expression as a difference of squares: \((2x)^2 - (3y)^2\).

Step 3: Apply the Difference of Squares Formula

Use the formula for the difference of squares, \(a^2 - b^2 = (a + b)(a - b)\), to factor the expression: \( (2x + 3y)(2x - 3y)\).

Key Concepts

Difference of Squares Formula

The difference of squares formula is a fundamental tool in algebra, particularly when working with polynomial expressions. This formula states that for any two terms a and b, the expression a² - b² can be factored into the product of two binomials, (a + b) and (a - b). Specifically, the formula is written as: \[ a^2 - b^2 = (a + b)(a - b) \].
When we come across an expression that looks like the difference between two squares, we can immediately use this formula to factor it. This is incredibly useful when simplifying expressions, solving equations, and performing operations in higher-level mathematics, such as calculus. For instance, the expression from our exercise, \(4x^2 - 9y^2\), can be identified as a difference of squares because it is comprised of two squared terms separated by a subtraction sign. By recognizing the structure as \(a^2 - b^2\) where \(a = 2x\) and \(b = 3y\), we can then apply the difference of squares formula to factor it into \((2x + 3y)(2x - 3y)\), simplifying the original expression effectively.

Factoring Polynomials

Factoring polynomials is much like breaking down numbers into their prime factors—it is the process of simplifying expressions into products of simpler ones. The polynomial expression \(4x^2 - 9y^2\) in the exercise is a type of quadratic polynomial. Factoring is often essential for solving polynomial equations and understanding the properties of graphs represented by those equations.
The strategy to factor depends on many factors, including the degree of the polynomial and the specific form it takes. For quadratic polynomials, common factoring techniques include using the greatest common factor (GCF), grouping, and utilizing special formulas like the difference of squares, which applies to our example. After determining that our expression is a difference of squares, the factoring process becomes straightforward and efficient, producing a set of factors that can offer insight into the roots or solutions to the related equation. It's worth noting that not all polynomials can be factored using integers or simple expressions; some require complex numbers or cannot be factored at all.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, operators, and variables. Operators include addition, subtraction, multiplication, division, and exponentiation, while variables represent unknown values and are usually denoted by letters such as x, y, or a. For example, in the expression \(4x^2 - 9y^2\), x and y are variables, 4 and 9 are coefficients, and the minus sign represents subtraction, a basic operation.
Algebraic expressions can range from simple, like 3x + 2, to complex, like polynomials and rational expressions.
Understanding how to work with and simplify these expressions is key to mastering algebra and advancing in mathematics. When we work with algebraic expressions, especially in factoring, we learn to recognize various patterns and structures, such as the difference of squares. This allows us to manipulate and simplify expressions to solve equations or simplify problems, as demonstrated in the step-by-step factorization of the given exercise.