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Chapter 2: Problem 12

Expand and combine like terms. $$ (x+7)(x-7) $$

Short Answer

Expert verified
Question: Expand and simplify the expression \((x+7)(x-7)\). Answer: The expanded and simplified expression is \(x^2 - 49\).

Step by step solution

01

Apply the FOIL method

First, we will apply the FOIL (First, Outer, Inner, Last) method to expand the expression. The FOIL method is a short way to remember how to multiply two binomials. Here, First terms: x * x Outer terms: x * (-7) Inner terms: 7 * x Last terms: 7 * (-7)
02

Multiply the terms

Now, multiply the terms found in step 1. First terms: x * x = x^2 Outer terms: x * (-7) = -7x Inner terms: 7 * x = 7x Last terms: 7 * (-7) = -49
03

Combine like terms

Combine the like terms (if any) found in step 2. x^2 - 7x + 7x - 49 Here, -7x and 7x are like terms, therefore we can combine them: x^2 - 49
04

Write the final result

The expanded and simplified form of the given expression \((x+7)(x-7)\) is: $$ x^2 - 49 $$

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Key Concepts

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

FOIL Method
When expanding two binomials, the FOIL method is a reliable tool. FOIL stands for First, Outer, Inner, Last, and it guides you through the multiplication process. For the expression
  • **First**: Multiply the first terms in each binomial, here it's \(x \times x = x^2\).
  • **Outer**: Next, multiply the outer terms: \(x \times (-7) = -7x\).
  • **Inner**: Then, multiply the inner terms: \(7 \times x = 7x\).
  • **Last**: Finally, multiply the last terms: \(7 \times (-7) = -49\).
Using the FOIL method simplifies the expansion, making it efficient and organized. Remembering the acronym can help you tackle similar problems with confidence.
Binomials
A binomial is an expression containing two terms, which are typically separated by a plus or minus sign. The expression \((x+7)\) and \((x-7)\) are examples of binomials.
Each binomial has two components: a term with a variable, and a constant term. In the given exercise, \(x\) is the variable and \(7\) and \(-7\) are constants. When multiplying binomials, you are essentially distributing each term in the first binomial across each term in the second.
This distribution is why the FOIL method is so useful: it helps break down the multiplication into manageable parts. Understanding binomials and how to manipulate them is key to mastering algebraic expansions.
Combining Like Terms
After expanding expressions, combining like terms is a crucial step to simplify the result. Like terms are terms that have the same variable raised to the same power. In the expression \(x^2 - 7x + 7x - 49\), the \(-7x\) and \(7x\) are like terms because they both contain the variable \(x\).
Combining them results in simplification: \(-7x + 7x = 0\). This leaves us with \(x^2 - 49\) as the simplified expression.
This process of combining improves the clarity and conciseness of the expression. Recognizing and merging like terms is a fundamental part of algebra that makes complex expressions more manageable.

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