/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Find each product. $$ (3 x-8... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 52

Find each product. $$ (3 x-8)(3 x+8) $$

Short Answer

Expert verified
The product is 9x^2 - 64.

Step by step solution

01

Identify the formula

Notice that the given expression (3x - 8)(3x + 8) is in the form of (a - b)(a + b), where a = 3x and b = 8. We can use the difference of squares formula: (a - b)(a + b) = a^2 - b^2.
02

Apply the difference of squares formula

Using the formula, (a - b)(a + b) = a^2 - b^2, substitute the values of a and b: (3x - 8)(3x + 8) = (3x)^2 - 8^2.
03

Calculate the squares

Now, compute the squares for each term: (3x)^2 = 9x^2 and 8^2 = 64. So, (3x - 8)(3x + 8) = 9x^2 - 64.
04

Combine the results

Combine the results to get the final product: (3x - 8)(3x + 8) = 9x^2 - 64.

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Key Concepts

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

algebraic expressions
Understanding algebraic expressions is crucial in algebra. An algebraic expression is a combination of numbers, variables, and arithmetical operations like addition, subtraction, multiplication, and division. For example, in our exercise, the expression (3x - 8)(3x + 8) consists of variables and constants. These are joined by multiplication and addition/subtraction signs. Here, 3x is a term where 3 is the coefficient and x is the variable.
factoring
Factoring is the process of breaking down an expression into simpler components that can be multiplied together to produce the original expression. In our exercise, we recognize that (3x - 8)(3x + 8) fits the difference of squares pattern, (a - b)(a + b), where a = 3x and b = 8. This simplifies to a^2 - b^2. Understanding this pattern helps us factor expressions efficiently.
polynomial multiplication
Polynomial multiplication involves multiplying each term in one polynomial by each term in another. For larger expressions, we often use the distributive property. In our specific problem, however, the expression fits a special formula called the difference of squares formula. Rather than multiplying every term, we use this formula to directly get a^2 - b^2. For (3x - 8)(3x + 8), we apply (3x)^2 - 8^2, which simplifies the multiplication step.
simplifying expressions
Simplifying expressions means rewriting them in a simpler or more concise form. From the previous steps, we found (3x - 8)(3x + 8) simplifies to (3x)^2 - 8^2. Next, we compute these squares: (3x)^2 = 9x^2 and 8^2 = 64. Subtracting these gives us the simplified form 9x^2 - 64. This final result is easier to work with and understand compared to the original more complex expression.

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