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Chapter 4: Problem 54

Find each product. $$ (6 y-3)(6 y+3) $$

Short Answer

Expert verified
36y^2 - 9

Step by step solution

01

Identify the Terms

The given expression is (6y - 3)(6y + 3). Notice that this expression is in the form of (a - b)(a + b).
02

Apply the Difference of Squares Formula

Use the formula for the difference of squares: (a-b)(a+b) = a^2 - b^2. Here, identify a = 6y and b = 3.
03

Substitute and Simplify

Substitute the values into the formula: a^2 - b^2 = (6y)^2 - 3^2. Simplify each term to get: (6y)^2 = 36y^2 and 3^2 = 9. So, (6y - 3)(6y + 3) = 36y^2 - 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product of Binomials
When we talk about the product of binomials, we refer to multiplying two binomial expressions. A binomial is an algebraic expression that contains exactly two terms, such as \( (x + 3) \) or \( (6y - 3) \). In our exercise, we have the binomials \( (6y - 3) \) and \( (6y + 3) \).

To find the product of these binomials, a useful approach is to identify patterns like the difference of squares formula. The formula for the difference of squares is:

\[ (a - b)(a + b) = a^2 - b^2 \]

This formula simplifies the multiplication process by reducing it to a subtraction of squares. Recognizing such patterns helps in quicker and easier calculations, as shown in the original solution.
Algebraic Expressions
An algebraic expression is a combination of variables, constants, and arithmetic operations. They form the basic building blocks of algebra. For example, \( 6y - 3 \) and \( 6y + 3 \) are both algebraic expressions because they consist of a variable \( y \), coefficients (6), and a constant term (3).

Understanding how to manipulate algebraic expressions is crucial. When dealing with multiplication of such expressions, identifying like terms and utilizing formulas can simplify the work significantly. In the given exercise, each term from the binomials has been carefully applied to the difference of squares formula, resulting in a streamlined multiplication process.
Simplification of Expressions
Simplification of expressions involves reducing them to their simplest form. For instance, in the solution, after recognizing the form (a-b)(a+b), we use the difference of squares formula:

\[ (6y - 3)(6y + 3) = (6y)^2 - 3^2 \]

Next, we perform the squaring:

\[ (6y)^2 = 36y^2 \]
\[ 3^2 = 9 \]

Finally, we substitute back to get:

\[ 36y^2 - 9 \]

By following these steps, the complex-looking product simplifies to a neat expression. Simplification makes understanding and further calculations more manageable, ensuring you can handle more complex algebraic operations efficiently.

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