Problem 10 - Elementary Algebra

Question

Factor completely. $$ 25 x 2-4 $$

Step-by-Step Solution

Answer

The expression factors to \((5x - 2)(5x + 2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the given expression \(25x^2 - 4\). In this case, the GCF is 1, as there are no common factors between 25 and 4 or involving the variable \(x\).

Step 2: Recognize the Expression as a Difference of Squares

The expression \(25x^2 - 4\) can be seen as a difference of squares. Recall that \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a^2 = (5x)^2\) and \(b^2 = 2^2\).

Step 3: Apply the Difference of Squares Formula

Now substitute \(a = 5x\) and \(b = 2\) into the difference of squares formula: \[(5x)^2 - 2^2 = (5x - 2)(5x + 2)\].

Step 4: Verify the Factored Expression

Multiply \((5x - 2)(5x + 2)\) to verify it results in the original expression \(25x^2 - 4\). Expanding gives \( (5x)(5x) + (5x)(2) - (2)(5x) - (2)(2) = 25x^2 - 4 \). This confirms the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest number or algebraic expression that can exactly divide each term of a polynomial. When factoring expressions, determining the GCF is a crucial first step because it simplifies the problem by reducing it to its simplest form.
For example, when examining the expression \(25x^2 - 4\), we quickly observe that there are no common numeric factors between the numbers 25 and 4. Moreover, the term \(x\) is present only in the first part of the expression, further limiting the GCF to 1.
Thus, when no common factors apart from 1 are present, the expression can still be simplified using other techniques such as identifying special patterns.

Difference of Squares

The concept of the difference of squares is a special pattern in algebra characterized by the expression \(a^2 - b^2\). This can be rewritten as \((a - b)(a + b)\). Identifying this pattern allows us to factor expressions quickly and efficiently.
In the case of \(25x^2 - 4\), recognize it as a difference of two squares because:

• The first term, \(25x^2\), is \((5x)^2\).
• The second term, \(4\), is \(2^2\).

Using the difference of squares formula, substitute \(a = 5x\) and \(b = 2\) to achieve the factorized form: \((5x - 2)(5x + 2)\).
This pattern is potent because it simplifies many algebraic expressions that would otherwise seem complex at first glance.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants organized through operations such as addition, subtraction, multiplication, and division. When expressions need to be simplified or factored, understanding their structure is key.
In our example, \(25x^2 - 4\), we have:

• A term, \(25x^2\), that involves a squared variable \(x\).
• A constant term, \(4\), which is numeric and stands alone without any variable.

Both terms are connected through subtraction, forming a clear distinction that enables techniques, like the difference of squares, to be employed effectively.
Recognizing various types of algebraic expressions and their standard factored forms provides a solid foundation for tackling more complex algebraic tasks.