Question

Factor \(16 x^2-81 y^2\) $$ (\quad)(\quad) $$ The following question contains a blank marked \begin{tabular}{|l|l} \hline Select... & Beneath the blank is a set of choices. \end{tabular} Indicate the choice that is correct and belongs in the blank. (Note: On the real GED" test, the choices will appear as a "drop-down" menu. When you click on a choice, it will appear in the blank.)

Short answer

\((4x + 9y)(4x - 9y)\)

Step-by-step solution

- Recognize the form of the expression

The given expression, \(16x^2 - 81y^2\), is in the form of a difference of squares. The general formula for factoring a difference of squares is \(a^2 - b^2 = (a + b)(a - b)\).

- Identify the squares

To apply the difference of squares formula, first identify the squares in the expression. Notice that \(16x^2\) is \((4x)^2\) and \(81y^2\) is \((9y)^2\).

- Apply the difference of squares formula

Using the form \(a^2 - b^2 = (a + b)(a - b)\), substitute \(a = 4x\) and \(b = 9y\). So, \(16x^2 - 81y^2\) can be written as \((4x + 9y)(4x - 9y)\).

- Write the final factored form

The expression \(16x^2 - 81y^2\) is factored as \((4x + 9y)(4x - 9y)\). Fill in the blanks with this factored form.

Key concepts

Difference of Squares

The given expression, \(16x^2 - 81y^2\), can be factored using the concept of the difference of squares. This is a common technique in algebra and is essential for your GED test preparation. The difference of squares formula is \[a^2 - b^2 = (a + b)(a - b)\]. This formula shows that any expression that looks like a square minus another square can be broken down into two binomials.
In the problem, recognize that \(16x^2\) is the square of \(4x\) and \(81y^2\) is the square of \(9y\). We substitute these into the formula:
1. Identify the squares: \(a = 4x\) and \(b = 9y\).
2. Apply the difference of squares formula: \(16x^2 - 81y^2 = (4x + 9y)(4x - 9y)\).
This gives the final factored form \((4x + 9y)(4x - 9y)\).

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this example, factoring polynomials is a critical skill. It involves breaking down a polynomial into simpler terms (factors) that, when multiplied together, give the original polynomial.
Understanding algebra helps in recognizing patterns and forms, such as the difference of squares. These patterns simplify problems, making them easier to solve. Here, the algebraic manipulation begins by recognizing a familiar form \(a^2 - b^2\).
Steps in algebra:

• Identify forms like squares and cubes
• Use established formulas
• Simplify expressions step by step
• Apply rules consistently

This step-by-step approach is vital not just for factoring but for solving any algebraic problem.

GED Test Preparation

For the GED test, mastering basic algebra skills like factoring is essential. The GED focuses on practical math skills, often presenting problems in real-world contexts. Understanding how to factor polynomials prepares you for various question types, whether direct algebra problems or applied math scenarios.
On the test, the problems might be multiple choice or require you to fill in the blanks, as seen in the exercise. Familiarize yourself with common algebraic forms and practice various problems.
Effective preparation tips:

• Practice regularly with a variety of algebra problems
• Understand each concept before moving on
• Use resources like textbooks and online platforms
• Take timed practice tests to simulate exam conditions

Consistent practice and thorough understanding will boost your confidence and competence in tackling GED math questions.