Problem 7 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$25 x^{2} y^{2}-36$$

Step-by-Step Solution

Answer

The expression \(25x^2y^2 - 36\) factors to \((5xy + 6)(5xy - 6)\).

Step 1: Identify the form

The given expression is \(25x^2y^2 - 36\). In order to use the difference of squares, we need to identify the terms as perfect squares. A difference of squares has the form \(a^2 - b^2\).

Step 2: Express each term as a square

First, express 25 as \((5)^2\) and \(x^2y^2\) as \((xy)^2\). So, \(25x^2y^2\) can be rewritten as \((5xy)^2\). Next, express 36 as \((6)^2\).

Step 3: Apply the difference of squares formula

The expression is now \((5xy)^2 - (6)^2\). Use the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\), where \(a = 5xy\) and \(b = 6\).

Step 4: Write the factored form

Substitute \(a\) and \(b\) into the formula: \((5xy + 6)(5xy - 6)\).

Step 5: Verify the factorization

Expand \((5xy + 6)(5xy - 6)\) to ensure it equals the original expression. The expanded form is \((5xy)^2 - (6)^2 = 25x^2y^2 - 36\), confirming the factorization is correct.

Key Concepts

Factoring Techniques

Factoring techniques are essential tools when dealing with algebraic expressions. They allow us to express a complex equation in a simpler form, making it easier to solve. One of the most popular techniques is the difference of squares. This method is specifically used when you have a subtraction of two perfect squares.
The formula for the difference of squares is:

• If you have an expression in the form of \(a^2 - b^2\), it can be factored into \((a + b)(a - b)\).

The beauty of this technique lies in its simplicity and power. By breaking down an expression into its simplest components, you can understand and manipulate it more effectively.
In the exercise given, we turned \(25x^2y^2 - 36\) into a difference of squares form. Recognizing \(25x^2y^2\) as \((5xy)^2\) and \(36\) as \(6^2\), we applied the difference of squares formula to factor it as \((5xy + 6)(5xy - 6)\). This technique makes even challenging quadratic expressions more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They form the basis for creating equations that you can solve. In understanding these expressions, it is crucial to recognize different forms and structures, such as the difference of squares.
To efficiently manipulate algebraic expressions, you need to view each term critically, identifying pairs like perfect squares which can be factored.

• Algebraic expressions often require rearrangement or simplification.
• Factoring techniques streamline solving these expressions.

Consider our exercise with \(25x^2y^2 - 36\). Recognizing this as an algebraic expression involving multiplication and subtraction is the first step. By identifying it as a difference of squares, we gain a more straightforward path to factorization. This process emphasizes the importance of identifying structural patterns to simplify and solve complex problems.

Quadratic Expressions

Quadratic expressions are specific types of algebraic expressions where the highest power of the variable is two. They often take the form \(ax^2 + bx + c\). However, when they appear in a difference of squares format, they can be solved very efficiently.
Quadratics, like our expression \(25x^2y^2 - 36\), ideally fit the difference of squares method.

• The key is to recognize perfect squares within the terms, like \((5xy)^2\) and \(6^2\).
• By rewriting these in the \(a^2 - b^2\) form, quadratic expressions become straightforward to factor.

In any quadratic expression problem, the primary goal is to simplify and solve it, often finding the roots or factors. Techniques such as factoring not only aid in solving but also in understanding the expression's overall behavior. Simplifying this involves using known patterns, like the difference of squares, to enhance your algebraic toolkit.