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Chapter 5: Problem 5

Find the greatest common factor for each list of terms. $$ 9 m^{3}, 3 m^{2}, 15 m $$

Short Answer

Expert verified
The GCF is 3m.

Step by step solution

01

- Identify the constants

Look at the numerical coefficients of each term: 9, 3, and 15. Identify the greatest common factor (GCF) of these numbers.
02

- Find the GCF of the constants

The factors of 9 are 1, 3, and 9. The factors of 3 are 1 and 3. The factors of 15 are 1, 3, 5, and 15. The greatest common factor of 9, 3, and 15 is 3.
03

- Identify the variables and their exponents

For the variables, look at the powers of the variable m: They are 3, 2, and 1.
04

- Find the lowest exponent

To find the greatest common factor for the variables, choose the lowest exponent. The smallest power of m is 1 (which is just m).
05

- Combine the results

Combine the results from the constants and the variables. The GCF of the constants is 3 and for the variables it is m. Thus, the GCF of the terms is 3m.

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Key Concepts

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

GCF
The Greatest Common Factor (GCF) is the largest number that can evenly divide two or more numbers. To find the GCF, you start by identifying all the factors of the given numbers.
For example, with the numbers 9, 3, and 15:
- The factors of 9 are: 1, 3, 9
- The factors of 3 are: 1, 3
- The factors of 15 are: 1, 3, 5, 15
The largest common factor among these is 3. Therefore, the GCF of 9, 3, and 15 is 3.
This method is essential in algebra when simplifying terms, as it helps reduce expressions to their simplest form.
Algebraic Terms
Algebraic terms consist of coefficients, variables, and exponents. Consider the terms: 9m^3, 3m^2, and 15m. Here, the coefficient is the numerical part of the term (9, 3, 15), while the variable part is the letter (m) along with its exponent (3, 2, 1).
Each term represents a product of these components. Understanding algebraic terms is crucial to performing operations like addition, subtraction, and factorization in algebra.
When simplifying or factoring expressions, we work with both the coefficients and the variable parts.
Factorization
Factorization is the process of breaking down an expression into products of its factors. This helps simplify expressions and solve equations efficiently.
In our example, to factorize the terms 9m^3, 3m^2, and 15m, we need to find the GCF of both the coefficients and the variables.
We've identified the GCF of the coefficients as 3. For the variables, we pick the smallest exponent, which is 1 (m). Combining these, we get the GCF as 3m.
Factorizing these terms by 3m, we get:
- 9m^3 梅 3m = 3m^2
- 3m^2 梅 3m = m
- 15m 梅 3m = 5
So, the original terms can be expressed as: 3m (3m^2 + m + 5). This makes complex expressions easier to handle and solve.

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