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Chapter 5: Problem 85

Find the GCF of each pair of monomials. $$5 a b, 8 b$$

Short Answer

Expert verified
The GCF of the monomials \(5ab\) and \(8b\) is \(b\).

Step by step solution

01

Determine the Factors of Both Monomials

List all the factors of each monomial separately. For the monomial \(5ab\), the factors are \(5\), \(a\), and \(b\). For the monomial \(8b\), the factors are \(8\) and \(b\).
02

Identify Common Factors

Compare the factors of both monomials to determine the common factors. The only factor that both \(5ab\) and \(8b\) share is \(b\).
03

Conclusion of the Greatest Common Factor

Since the only common factor is \(b\), the greatest common factor (GCF) of the monomials \(5ab\) and \(8b\) is \(b\).

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Key Concepts

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

Understanding Monomials
Monomials are algebraic expressions that consist of only one term. They can include numbers, variables, or both multiplied together. For example, in the exercise provided, you have monomials like \(5ab\) and \(8b\).
Monomials are important because they help simplify and solve algebraic problems.
  • A monomial like \(5ab\) consists of a coefficient (5) and variables \(a\) and \(b\).
  • In \(8b\), 8 is the coefficient and \(b\) is the variable.
It's essential to clearly identify all parts of a monomial to effectively perform operations such as factoring or finding the greatest common factor (GCF). Once monomials are fully understood, working with complex algebraic expressions becomes a lot easier.
Factoring Monomials
Factoring is the process of breaking down a larger expression into a product of simpler components, known as factors. When factoring monomials, you aim to express them as a product of their basic parts. Take monomials like \(5ab\) and \(8b\) for example.
The goal of factoring these monomials is to identify all their possible factors.
  • For \(5ab\), the factors are \(5\), \(a\), and \(b\).
  • In the case of \(8b\), the factors are \(8\) and \(b\).
Factoring plays a crucial role when finding the GCF because it helps pinpoint which factors are shared between monomials.
By breaking down monomials into their simplest forms, factoring simplifies the process of solving algebraic expressions.
Identifying Common Factors
Common factors are factors that are shared between two or more numbers or expressions. They are crucial in finding the greatest common factor (GCF). In the given exercise, finding common factors is the key step to solving the problem.
When you list out the factors of each monomial, you look for similarities.
  • The factors of \(5ab\) are \(5\), \(a\), and \(b\).
  • Similarly, the factors of \(8b\) are \(8\) and \(b\).
In this case, you can see the common factor is \(b\) since both monomials share it.
Recognizing common factors among monomials allows you to determine the GCF. This helps in simplifying expressions and solving equations more efficiently.

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