Problem 47 - Elementary Algebra

Question

Factor completely. $$ 4 x 2-49 $$

Step-by-Step Solution

Answer

The expression factors to \((2x - 7)(2x + 7)\).

Step 1: Recognize the Expression Type

The given expression is of the form \(4x^2 - 49\). This is a difference of squares because it can be expressed as \((2x)^2 - 7^2\).

Step 2: Apply the Difference of Squares Formula

We use the difference of squares formula: \[a^2 - b^2 = (a - b)(a + b)\]In our case, \(a = 2x\) and \(b = 7\).

Step 3: Factor the Expression Using the Formula

Substitute \(a = 2x\) and \(b = 7\) into the difference of squares formula: \[(2x - 7)(2x + 7)\]This is the completely factored form of the expression.

Key Concepts

Difference of Squares

The difference of squares is a special pattern or identity used to factor certain polynomials. It applies when you have an expression of the form \(a^2 - b^2\), which can be thought of as "the square of something minus the square of something else." In our example, we have \(4x^2 - 49\). Notice we can rewrite this as \((2x)^2 - 7^2\).

To factor the difference of squares, we use the identity: \(a^2 - b^2 = (a - b)(a + b)\). This identity reveals that the expression can be split into two binomials.

• One binomial is the difference of the terms \(a\) and \(b\).
• The other binomial is their sum.

This pattern simplifies the expression into a pair of linear factors, making it easier to work with in equations or further algebraic operations.

Expression Types

Expressions in algebra can come in different forms, and identifying these forms is crucial to choosing the correct factoring technique. The expression \(4x^2 - 49\) is an example of a specific expression type known as a 'difference of squares.'

Recognizing an expression type involves looking for specific patterns:

• Binomial: An expression with two terms, like \(a^2 - b^2\).
• Squares: Check if both terms are perfect squares.
• Difference: There is a subtraction between the two squared terms.

For our expression \((2x)^2 - 7^2\), it fits beautifully with the difference of squares formula due to its structure. Recognizing this allows us to promptly apply the right method to factor the polynomial completely. Identifying expression types accurately helps streamline the factoring process and minimizes errors.

Factoring Techniques

Factoring techniques are methods we use to rewrite expressions down into simpler pieces called "factors." The goal of factoring is to simplify or solve polynomial expressions. Different types of polynomials require different techniques to be effectively factored.

The difference of squares is one such technique. It comes under broader techniques that include:

• Greatest Common Factor (GCF): Pull out the largest common factor from all terms.
• Trinomial Factoring: For expressions like \(ax^2 + bx + c\).
• Grouping: Used when an expression has four or more terms.

When you spot the difference of squares, like in our exercise \((2x)^2 - 7^2\), you apply its specific formula \(a^2 - b^2 = (a - b)(a + b)\). Each technique is a tool neatly tailored for certain expression types, making algebraic manipulation not only possible but manageable and efficient. By understanding and practicing these techniques, you'll find factoring becomes an easier and automatic part of working with polynomials.