Problem 21 - Algebra and Trigonometry

Question

\(21-28\) Use a Factoring Formula to factor the expression. $$ 9 a^{2}-16 $$

Step-by-Step Solution

Answer

The expression factors to \((3a + 4)(3a - 4)\).

Step 1: Identify the Factoring Type

The expression is a difference of squares. It is in the form \( a^2 - b^2 \).

Step 2: Apply the Factoring Formula

The difference of squares formula is \( a^2 - b^2 = (a + b)(a - b) \).

Step 3: Express Given Expression as a Difference of Squares

For the expression \(9a^2 - 16\), recognize it as \((3a)^2 - (4)^2\).

Step 4: Factor the Expression

Using the difference of squares formula, factor \(9a^2 - 16\) as \((3a + 4)(3a - 4)\).

Key Concepts

Difference of Squares

When dealing with polynomials, recognizing patterns helps simplify expressions. One of the common patterns is the "difference of squares". Let's break this down.

The "difference of squares" occurs when an expression is in the form \( a^2 - b^2 \). Here, "difference" implies subtraction, and "squares" indicates that each term is a perfect square.

Key characteristics of the difference of squares include:

• There are two terms.
• The terms are separated by a minus sign.
• Each term is a perfect square, which means it can be expressed as another expression squared.

A classic example is \( x^2 - 4 \), which can be rewritten as \( (x)^2 - (2)^2 \).

This recognition allows us to use specific factoring techniques, simplifying calculations and solving problems more efficiently.

Factoring Formulas

Factoring is a method used to express an equation or expression as a product of its factors. In the case of the difference of squares, a specific factoring formula applies.

The formula for factoring a difference of squares is:

• \( a^2 - b^2 = (a + b)(a - b) \)

This formula is a powerful tool in algebra, allowing us to break down complex expressions into simpler components.

Why is this useful? Because it reduces seemingly complex problems into a product of straightforward factors, making it easier to view relationships and simplify expressions.

Take our original expression \( 9a^2 - 16 \). Recognizing this as \( (3a)^2 - (4)^2 \), applying the factoring formula gives us \( (3a + 4)(3a - 4) \).

By mastering this formula, you can handle a wide range of expressions in algebra, simplifying and solving them efficiently.

Algebraic Expressions

At the heart of algebra is the ability to work with algebraic expressions. These expressions contain numbers, variables, and operations, forming the building blocks of equations and functions.

Working with algebraic expressions involves several tasks:

• Simplifying expressions by combining like terms or using properties such as the distributive property.
• Simplifying by factoring to reveal the products of factors.
• Understanding the relationships between numbers and variables, which can be linear, quadratic, or another form depending on degree and terms.

In the case of \( 9a^2 - 16 \), recognizing it as an algebraic expression composed of two squared terms is the first step.

This understanding allows us to utilize techniques like the difference of squares to break it into its factors \( (3a + 4)(3a - 4) \).

Algebraic expressions are at the core of much of algebra, offering a foundation for solving problems and understanding mathematical relationships.