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Chapter 5: Problem 50

Multiply. $$ (a-3 b)(a+3 b) $$

Short Answer

Expert verified
The product is \(a^2 - 9b^2 \).

Step by step solution

01

Identify the distributive property

Recognize that the expression \( (a-3b)(a+3b) \) can be expanded using the distributive property of algebra, also known as the FOIL method: First, Outer, Inner, Last.
02

Apply the FOIL method

Multiply the terms: \(a \times a \) for First, \(-3b \times a \) for Outer, \(a \times 3b \) for Inner, and \(-3b \times 3b \) for Last.
03

Perform each multiplication

Calculate each part: \(a \times a = a^2\), \(-3b \times a = -3ab\), \(a \times 3b = 3ab\), and \(-3b \times 3b = -9b^2\).
04

Combine like terms

Sum the results: \(a^2 - 3ab + 3ab - 9b^2 \). Notice that \(-3ab + 3ab \) cancels out.
05

Simplify the expression

Combine and simplify the result to get the final answer: \(a^2 - 9b^2 \).

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Key Concepts

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

distributive property
The distributive property is a fundamental principle in algebra. It allows you to multiply a single term by each term within a parenthesis. In the given problem, we start with the expression \( (a-3b)(a+3b) \). Using the distributive property, we break it down into smaller, manageable parts. This helps us systematically expand and simplify complex expressions using simpler multiplications.
FOIL method
The FOIL method is a technique specifically used for multiplying two binomials. It stands for:
  • First: Multiply the first terms in each binomial (\(a \times a = a^2\))
  • Outer: Multiply the outer terms (\(-3b \times a = -3ab\))
  • Inner: Multiply the inner terms (\(a \times 3b = 3ab\))
  • Last: Multiply the last terms in each binomial (\(-3b \times 3b = -9b^2\))

By following this method, we ensure that no terms are missed, and it allows us to systematically approach the multiplication of binomials like \( (a-3b)(a+3b) \).
like terms
Like terms are terms that have the same variable raised to the same power. In our expression, we find terms like \(-3ab\) and \(3ab\). These terms have identical variables and exponents, which means they can be combined.
When we sum the results: \(a^2 - 3ab + 3ab - 9b^2\), the like terms \(-3ab + 3ab\) cancel each other out, leaving us with the simplified expression \(a^2 - 9b^2\). Recognizing and combining like terms simplifies algebraic expressions, making them easier to work with.

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