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Chapter 4: Problem 63

Factor each monomial. $$9 n$$

Short Answer

Expert verified
The factors of the monomial \(9n\) are \(3\), \(3\), and \(n\).

Step by step solution

01

Identify the Coefficient and Variable

The monomial given is \(9n\). Here, \(9\) is the coefficient and \(n\) is the variable that needs to be factored.
02

Find Factors of the Coefficient

The coefficient \(9\) can be factored into its prime factors. The prime factorization of \(9\) is \(3 \times 3\).
03

Write the Factored Form

Combine these factors with the variable \(n\) to express the monomial in a factored form. This forms \((3 \times 3) \times n\), which simplifies to \(3 \times 3 \times n\). You do not need to pull out any further factors because \(n\) is already in its simplest form.

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Key Concepts

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

Prime Factorization
Prime factorization is a technique used to express a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. In the context of factoring a monomial, breaking down the coefficient of the monomial into its prime factors helps understand how the monomial is built.

For instance, the coefficient in the monomial \(9n\) is 9. It can be break down into smaller parts using the prime factorization method:
  • Start with the smallest prime number, 2. Since 9 is not even, try the next prime number.
  • The next prime is 3, which divides 9 to give 3.
  • The result can be divided by 3 again, giving 1.
Hence, the prime factorization of 9 is \( 3 \times 3 \). This means 9 is made up of multiplying two 3s together.
Coefficients
In mathematical terms, a coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression. It's like the driving force of the term.

Looking at the monomial \(9n\), the number 9 is the coefficient. It tells us how many times the variable \(n\) is considered in this term. Imagine coefficients as a group of products required to complete a project, with the variable being the task. If you have a coefficient of 9, it means you need 9 of whatever the variable, or task, represents.
  • A simple coefficient like 9 can affect the value, magnitude, or direction (if negative) of the expression it is in.
  • Understanding coefficients is essential, especially when working with larger expressions and equations, as they lay the groundwork for solving them effectively.
Variables
Variables represent unknown values and are usually denoted by letters, like \(n\) in the given monomial \(9n\). They stand in place of numbers that we often do not know initially.

Think of variables as placeholders or labels in a programming context. They hold or refer to the data that may change within a given problem. When factoring a monomial such as \(9n\):
  • The variable \(n\) is a critical component as it suggests that this term is part of something larger where the exact value of \(n\) may vary.
  • Variables play a substantial role in algebra, enabling the simplification of mathematical expressions and solving equations.
By factoring the monomial \(9n\), you represent it as a product of its parts: \(3 \times 3 \times n\), which shows how the coefficient and variable work together to form the expression.

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