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Chapter 6: Problem 86

Multiply. $$ (2 n+7)(2 n-7) $$

Short Answer

Expert verified
4n^2 - 49

Step by step solution

01

- Recognize the Pattern

The expression \( (2n+7)(2n-7) \) fits the form of a difference of squares, which is \( (a+b)(a-b) = a^2 - b^2 \). Here, \( a = 2n \) and \( b = 7 \).
02

- Apply the Difference of Squares Formula

Use the pattern identified to simplify the expression: \( (2n+7)(2n-7) = (2n)^2 - 7^2 \).
03

- Calculate the Squares

Calculate \( (2n)^2 \) and \( 7^2 \). \( (2n)^2 = 4n^2 \) and \( 7^2 = 49 \).
04

- Simplify the Expression

Subtract the two results from Step 3: \( 4n^2 - 49 \).

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Key Concepts

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

Algebraic Patterns
Understanding algebraic patterns is key in tackling various algebra problems, including those involving the difference of squares. A common pattern is the difference of squares, which is seen in the expression \( (a + b)(a - b) \). This pattern tells us that the product of a sum and a difference of the same terms is equal to the difference of their squares. Recognizing such patterns helps simplify expressions and solve problems more efficiently. In our specific exercise, the pattern \( (2n + 7)(2n - 7) \) matches the form of \( (a + b)(a - b) \), allowing us to apply the formula directly.
Binomial Multiplication
Binomial multiplication involves multiplying two binomials, which can often be simplified using patterns like the difference of squares. In our exercise, we use the pattern to multiply \( (2n + 7)(2n - 7) \). Recognize that here, we don't need to use the FOIL (First, Outer, Inner, Last) method because the terms fit the difference of squares. The steps are:
  • Identify each part of the binomials, where \(a = 2n\) and \(b = 7\).
  • Apply the difference of squares formula: \( (a + b)(a - b) = a^2 - b^2 \).
  • Calculate the squares: \( (2n)^2 = 4n^2 \) and \( 7^2 = 49 \).
  • Subtract to get \( 4n^2 - 49 \).
This substitution and simplification make the problem straightforward.
Simplification of Expressions
Simplifying algebraic expressions makes them easier to work with and understand. Our exercise demonstrates this by using patterns and straightforward calculations. After recognizing the difference of squares pattern and applying it:
  • We simplified \( (2n + 7)(2n - 7) \) to \( (2n)^2 - 7^2 \).
  • We then calculated the squares: \( (2n)^2 = 4n^2 \) and \( 7^2 = 49 \).
  • Finally, we simplified the expression to \( 4n^2 - 49 \).
By breaking down the process into steps, it becomes clear how each component of the expression is handled, making the simplification straightforward. Understanding these steps is essential for mastering algebraic manipulations and tackling more complex problems.

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