Problem 18 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$16 s^{2}-(3 t+1)^{2}$$

Step-by-Step Solution

Answer

The factorization is \((4s + 3t + 1)(4s - 3t - 1)\).

Step 1: Recognize the Difference of Squares Pattern

The difference of squares pattern is defined as \( a^2 - b^2 = (a + b)(a - b) \). We need to identify expressions that can be represented in this form in the given equation: \( 16s^{2} - (3t+1)^2 \).

Step 2: Identify the Squares

In the expression \( 16s^{2} - (3t+1)^2 \), we identify \( 16s^{2} \) as \((4s)^2\), and \( (3t + 1)^2 \) as it is already expressed as a square. Thus, \( a = 4s \) and \( b = (3t+1) \).

Step 3: Apply the Difference of Squares Formula

Using the formula \( a^2 - b^2 = (a + b)(a - b) \), substitute \( a = 4s \) and \( b = (3t+1) \). This gives us \( (4s + (3t + 1))(4s - (3t + 1)) \).

Step 4: Simplify the Expression

Simplify the expression from Step 3: \( (4s + (3t + 1))(4s - (3t + 1)) \) becomes \( (4s + 3t + 1)(4s - 3t - 1) \).

Key Concepts

Factoring Algebraic Expressions

Understanding how to factor algebraic expressions is a crucial skill in algebra. It involves breaking down complex expressions into simpler ones, often in the form of a product of factors. Factoring simplifies expressions, making it easier to solve equations and find out important properties, like roots or intercepts. In the problem of factoring the expression \(16s^{2} - (3t+1)^2\), the goal is to express it in a factored form using the difference of squares pattern.

To efficiently factor, follow these steps:

• Identify any potential squares in the expression.
• Recognize patterns such as the difference of squares.
• Apply algebraic identities (like the difference of squares) to simplify the expression.

By mastering these steps, you can tackle a wide variety of algebraic expressions and easily factor them into their simpler components.

Squared Terms Identification

Identifying squared terms is a fundamental aspect of recognizing patterns in algebraic expressions. Squared terms are often written in the format \(a^2\) and are crucial when using formulas like the difference of squares. In the equation given, \(16s^{2} - (3t+1)^2\), it is important to express each part as a perfect square.

Here’s how you identify them:

• Recognize that \(16s^{2}\) is a square term, equivalent to \((4s)^2\).
• The term \((3t+1)^2\) is already expressed as a square, where \(b = (3t+1)\).

By clearly identifying the squared terms, you can simplify each equation step and quickly see the path to the factoring step. Don't forget: breaking down complicated parts into recognizable squares is key to simplifying further.

Algebraic Simplification

Algebraic simplification is the last step in turning complex expressions into simple, usable forms. After you've factored out the expression using the difference of squares pattern, you often need to simplify further. This involves an operation on expressions to make them more concise or to prepare them for solving or graphing.

In our expression \((4s + (3t + 1))(4s - (3t + 1))\), simplification involves:

• Removing parentheses by distributing any constant or variable across the sum or difference inside the parentheses.
• Reducing terms to their simplest form, like changing \((4s + 3t + 1)\) and \((4s - 3t - 1)\).

In this case, simplification resulted in \((4s + 3t + 1)(4s - 3t - 1)\). This step is vital as it provides a version of the expression that is often more functional and contributes to better understanding and usability in further mathematical applications.