Problem 106 - Algebra 2

Question

Factor each expression. $$ 16 a^{2}-9 b^{2} $$

Step-by-Step Solution

Answer

The factored expression is \((4a + 3b)(4a - 3b)\)

Step 1: Recognize the Format

The given expression is \(16a^{2} - 9b^{2}\). It is in the form of a difference of squares. This is recognized by the standard form \(a^{2} - b^{2}\).

Step 2: Determine the a and b Values

In this equation, \(16a^{2}\) is the \(a^{2}\) value and \(9b^{2}\) is the \(b^{2}\) value. So, \(a\) is \(4a\) (because \(4a × 4a =16a^{2}\)) and \(b\) is \(3b\) (because \(3b × 3b = 9b^{2}\)).

Step 3: Apply the Difference of Squares Formula

Apply the difference of squares formula \(a^{2} - b^{2} = (a + b)(a - b)\) to this problem by replacing \(a\) with \(4a\) and \(b\) with \(3b\). The factored form of the expression is then \((4a + 3b)(4a - 3b)\).

Key Concepts

Difference of Squares

The difference of squares is a crucial concept in algebra when it comes to factoring certain expressions. This entails a squared term subtracted from another squared term. Specifically, it takes the form \(a^2 - b^2\), where both \(a^2\) and \(b^2\) are perfect squares and can be expressed as products of the same term by itself.

• Key point: The expression must be a subtraction of two squares.
• Formula: \(a^2 - b^2 = (a + b)(a - b)\)
• Visualize it: Imagine you have a square plot with side \(a\), and you're subtracting a smaller square with side \(b\).

This formula simplifies the factoring process, breaking down the expression into a product of two binomials, making it easier to handle. Understanding this concept is vital because it frequently appears in algebraic problems, helping to simplify seemingly complex expressions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. In the expression \(16a^2 - 9b^2\), each term is an algebraic term that combines a coefficient (the number part) with a variable (the part involving letters like \(a\) and \(b\)).

• The term \(16a^2\) involves a coefficient of 16 and a variable that is squared: \(a^2\).
• The term \(9b^2\) has a coefficient of 9, with the variable squared: \(b^2\).

Algebraic expressions can be simple or complex, but understanding their components is essential for factoring. You break them down into their individual terms and apply various mathematical operations to simplify them. Recognizing patterns, such as the difference of squares within them, is a key skill to develop.

Factoring Techniques

Factoring is a method used to express an algebraic expression as a product of its factors. This is a fundamental skill in algebra that simplifies expressions and solves equations. The challenge is to rewrite expressions in a "factored form" where the expression is expressed as a multiplication of simpler expressions.One effective technique is using the difference of squares formula. To apply this method, you need to:

• Identify if the expression is a difference of squares; this form allows you to use the specific formula \(a^2 - b^2 = (a + b)(a - b)\).
• Recognize perfect square terms and find what squared numbers produced them.
• Apply these values into the formula to rewrite the expression as a product of two binomials.

Mastering different factoring techniques, like identifying the difference of squares, prepares you to tackle more complex algebraic problems with confidence. It's like unlocking a toolkit for solving equations and understanding how polynomials work.