Problem 54 - Beginning and Intermediate Algebra

Question

Factor completely. $$9 a^{2}-1$$

Step-by-Step Solution

Answer

The short answer is: \(9a^2 - 1 = (3a + 1)(3a - 1)\)

Step 1: Identify a and b

From the given expression \(9a^2 - 1\), let's identify a and b. Here, we have: \(a^2 = 9a^2\), so a = \(3a\) and \(b^2 = 1\), so b = 1 Now that we have identified our a and b, we can proceed to factor the given expression.

Step 2: Apply Difference of Squares Formula

To factor the expression, we will use the formula: \(a^2 - b^2 = (a + b)(a - b)\) Replace a with \(3a\) and b with 1: \((3a + 1)(3a - 1)\) The factored expression is: \((3a + 1)(3a - 1)\)

Key Concepts

Difference of Squares

The term "difference of squares" refers to a specific pattern in algebra used to simplify or factor expressions. It is quite common in polynomial factorization. This pattern involves an expression that comprises the difference between two squared terms, such as in the following standard form:

• \( a^2 - b^2 \)

There is a special formula for factoring such expressions:

• \( a^2 - b^2 = (a + b)(a - b) \)

This formula is powerful because it turns the original expression into a product of two binomials, making it easier to solve or simplify later on. In the original problem, \( 9a^2 - 1 \) can be viewed as \( (3a)^2 - 1^2 \), illustrating that it fits the difference of squares pattern.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the building blocks of many algebra problems and lead to the creation of equations and inequalities. For instance, an expression like \( 9a^2 - 1 \) consists of terms involving both coefficients and variables, specifically:

• The term \( 9a^2 \) includes a variable \( a \) raised to the power of 2 and multiplied by the coefficient 9.
• Similarly, \(-1\) is a constant term that doesn’t include a variable.

Understanding the structure of algebraic expressions is crucial because it helps identify what type of factorization or simplification approach is necessary, such as recognizing the difference of squares pattern in given problems.

Factor Completely

To "factor completely" means to express a polynomial as a product of simpler polynomials that cannot be factored further over the set of integers. The goal is to break down the expression fully using suitable factorization techniques.

• Starting with \( 9a^2 - 1 \), we first identify it as a difference of squares.
• Using the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \), we substitute \( a = 3a \) and \( b = 1 \).
• This leads to factoring the expression into \((3a + 1)(3a - 1)\).

Both of these binomials, \(3a + 1\) and \(3a - 1\), can't be factored further, signifying that you've factored the expression completely. Understanding this ensures you can simplify expressions effectively and solve algebraic problems with confidence.