Problem 21 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 9 c^{2}-1 $$

Step-by-Step Solution

Answer

The factorization of the given expression \(9c^{2} - 1\) is \((3c + 1)(3c - 1)\).

Step 1: Identify the Difference of Squares

Recognize that the expression \(9c^{2} - 1\) can be rewritten as \((3c)^{2} - (1)^{2}\). This shows that it is a difference of squares since it is in the form \(a^{2} - b^{2}\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states that \(a^{2} - b^{2} =(a+b)(a-b)\). Thus, let a = 3c and b = 1. Apply the formula gives the factorization as \((3c + 1)(3c - 1)\).

Key Concepts

Difference of Squares

The concept of the 'Difference of Squares' is an essential algebraic tool used in simplifying expressions. It applies to expressions where two perfect squares are subtracted. A perfect square is simply a number multiplied by itself. The general formula to remember for the difference of squares is:

• \(a^2 - b^2 = (a + b)(a - b)\)

To illustrate, let's look at the expression \(9c^2 - 1\). Notice that \(9c^2\) is a perfect square, as it can be rewritten as \((3c)^2\), and \(1\) is \(1^2\) which is also a perfect square.
Identifying these components allows for straightforward application of the formula. This type of factorization simplifies solving and graphing these equations, making it a popular method in both academic and real-world problem-solving. Recognizing a difference of squares can help simplify your work significantly. In our example, using this concept allows us to factor \(9c^2 - 1\) into \((3c + 1)(3c - 1)\).

Quadratic Expressions

Quadratic expressions are algebraic expressions of degree two, which means they involve the square of a variable. They generally take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.
In the exercise, \(9c^2 - 1\) is a specific kind of quadratic known as a binomial quadratic because it consists of two terms. An important aspect of dealing with quadratic expressions is identifying their structure, which can help in choosing the appropriate method for factoring. For example, when the quadratic expression is in the difference of squares form, applying the formula \((a+b)(a-b)\) becomes ideal.
Understanding the essentials of quadratic expressions aids you in predicting the values where the expression equals zero, known as roots or solutions. This ability is vital not only in math but also in fields like physics and engineering where predicting outcomes is necessary. This process of looking for perfect squares and their differences in a quadratic expression is exactly what we did with \(9c^2 - 1\).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to solve problems or make expressions more manageable. In our exercise, algebraic manipulation was used to transform \(9c^2 - 1\) into a product of its factors.
The key manipulation involved here is recognizing patterns - specifically the pattern of the difference of squares, and then applying the corresponding factorization formula. Moreover, renaming or redefining components, as seen with \(9c^2 = (3c)^2\), is critical during manipulation. It often involves combining like terms, distributing, and factoring, which are all necessary skills in algebra.
Once the structure of an expression is altered through algebraic manipulation, it becomes simpler to solve or graph. This skill is incredibly useful for algebra students as it forms the foundation for more advanced topics. Effective manipulation not only aids in finding solutions but also in understanding the behavior and characteristics of the function described by the expression. Mastering these techniques ensures confidence in handling diverse mathematical challenges.