Problem 61 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(4 x^{2}-25\)

Step-by-Step Solution

Answer

The expression \(4x^2 - 25\) factors to \((2x - 5)(2x + 5)\).

Step 1: Identify the expression as a special form

The given expression is \(4x^2 - 25\). Notice that it resembles the difference of squares, which is a special algebraic form \(a^2 - b^2\) that can be factored into \((a - b)(a + b)\).

Step 2: Express each term as a square

Express each term in the form of a square: \(4x^2=(2x)^2\) and \(25 = 5^2\). Thus, the expression becomes \((2x)^2 - 5^2\).

Step 3: Apply the difference of squares formula

Now that the expression \((2x)^2 - 5^2\) is in the difference of squares form \(a^2 - b^2\), we apply the formula: \((a-b)(a+b)\). Therefore, we get \((2x - 5)(2x + 5)\).

Step 4: Verify the factorization

Multiply \((2x - 5)\) and \((2x + 5)\) to check: \((2x)(2x) + (2x)(5) - (5)(2x) - (5)(5) = 4x^2 + 10x - 10x - 25 = 4x^2 - 25\). This confirms the factorization is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It involves rewriting a polynomial as a product of simpler polynomial expressions. This process simplifies complex expressions and is essential in solving polynomial equations. Typically, factoring starts by identifying any common factors in each term of the polynomial. Once common factors are factored out, the remaining expression can be further broken down into factors based on specific algorithms and techniques (like special algebraic forms such as the difference of squares).

Why do we factor polynomials? Here are some key reasons:

• To simplify algebraic expressions.
• To solve equations—factoring is often a required step in finding the roots of polynomial equations.
• To facilitate graphing—particularly useful for finding x-intercepts of polynomial graphs.

Understanding the role of each term in the polynomial, as well as potential patterns like those found in special forms, can make factoring a more intuitive and straightforward process.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (such as addition and multiplication). They are the building blocks of algebra, representing mathematical relationships and patterns. In our exercise example, the expression is a binomial (two terms): \(4x^2 - 25\).

When dealing with algebraic expressions, it's essential to familiarize yourself with:

• Terms—Each part of an algebraic expression separated by a plus or minus sign. In \(4x^2 - 25\), \(4x^2\) and \(25\) are the terms.
• Coefficients—The numerical part of terms involving variables. For the term \(4x^2\), 4 is the coefficient.
• Variables—Symbols that represent unspecified numerical values; \(x\) in our example.
• Constants—Numbers without variables; in the expression, 25 is a constant.

Understanding these components will greatly assist in manipulating and solving expressions, especially in factoring processes.

Special Algebraic Forms

Special algebraic forms like the difference of squares offer a shortcut in factoring. Recognizing these forms can save time and simplify the process of solving expressions. The common special forms you’ll encounter include:

• Difference of squares: Recognizable by expressions like \(a^2 - b^2\), and always factored as \((a - b)(a + b)\).
• Perfect square trinomials: Expressions in the form \((a + b)^2\) or \((a - b)^2\), which expand to \(a^2 + 2ab + b^2\) and \(a^2 - 2ab + b^2\) respectively.
• Sum and difference of cubes: Patterns like \(a^3 + b^3\) and \(a^3 - b^3\) have their special factorizations.

In the exercise, \(4x^2 - 25\) is recognized as a difference of squares because both \(4x^2\) and \(25\) are perfect squares, allowing us to factor it as \((2x - 5)(2x + 5)\). Developing the ability to spot and use these forms is a valuable skill in algebra, streamlining the solving process and revealing deeper mathematical insights.