Problem 28 - Elementary Algebra

Question

Factor completely. $$ a 2 b_{2}-36 $$

Step-by-Step Solution

Answer

The expression factors to \((ab - 6)(ab + 6)\).

Step 1: Identify the Greatest Common Factor (GCF)

The given expression is \( a^2b^2 - 36 \). First, identify the greatest common factor of the terms in the expression. In this case, there is no common factor between the two terms, so the GCF is 1.

Step 2: Apply the Difference of Squares Formula

Notice that the given expression \( a^2b^2 - 36 \) resembles the difference of squares form \( x^2 - y^2 = (x-y)(x+y) \). Identify \( x = ab \) and \( y = 6 \), since \( (ab)^2 = a^2b^2 \) and \( 6^2 = 36 \).

Step 3: Factor Using the Difference of Squares

Apply the difference of squares formula to factor the expression: \( a^2b^2 - 36 = (ab - 6)(ab + 6) \). This is the complete factorization of the expression.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial concept in algebraic factoring. It represents the largest factor shared by two or more terms in an algebraic expression. To find the GCF, list all the factors of each term and identify the greatest factor that appears in each list.

In some cases, like the expression given, there might not be a common factor other than 1. This can happen when the terms do not share any factors besides the number 1 itself. Despite this, always check for a GCF first, as it simplifies further operations. Recognizing the GCF is an essential first step before applying other factoring techniques, such as factoring by grouping or using special formulas.

Difference of Squares

The difference of squares is a specific technique used in polynomial factorization. This formula applies when you have an expression in the form of \( x^2 - y^2 \), where both terms are perfect squares. The formula to factor the difference of squares is given by:
- \( x^2 - y^2 = (x-y)(x+y) \).

To apply this formula, identify two terms that are squares and subtracting each other. For the expression \( a^2b^2 - 36 \), we notice that \( a^2b^2 \) can be rewritten as \( (ab)^2 \) and \( 36 \) is \( 6^2 \). Here, \( x \) is \( ab \) and \( y \) is 6. The expression fits the difference of squares format, allowing us to factor it as \( (ab - 6)(ab + 6) \).

Using the difference of squares formula is a powerful method to simplify quadratic expressions and should be part of your factoring toolbox.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler, non-divisible factors. Factoring is crucial because it allows you to understand the roots of the polynomial, simplify expressions, and solve equations.

There are several methods for factoring polynomials, including:

• Identifying the Greatest Common Factor (GCF)
• Difference of Squares and other special product formulas
• Trial and error, especially for trinomials
• Grouping terms to simplify the polynomial

Each method has its application and is chosen based on the polynomial's structure. In the example given, the polynomial \( a^2b^2 - 36 \) was factored using both the identification of the GCF (which was 1) and the difference of squares technique. Mastery of these methods makes handling more complex algebraic expressions much easier and more efficient.