Problem 35 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 6 m^{4} n, 12 m^{3} n^{2}, 9 m^{3} n^{3} $$

Step-by-Step Solution

Answer

The GCF is \(3m^3n\).

Step 1: List the Factors of Each Coefficient

Identify the factors of the coefficients in each term. The coefficients are 6, 12, and 9. - The factors of 6 are: 1, 2, 3, 6. - The factors of 12 are: 1, 2, 3, 4, 6, 12. - The factors of 9 are: 1, 3, 9.

Step 2: Identify the Common Factors

Determine the common factors from the lists of factors found in Step 1. The common factors of 6, 12, and 9 are 1 and 3.

Step 3: Choose the Greatest Common Factor for Coefficients

From the common factors identified, choose the greatest one. The greatest common factor of the coefficients is 3.

Step 4: List the Variables with the Lowest Powers

Consider the variables in each term. For the terms given: - All terms have both variables \(m\) and \(n\). - The powers of \(m\) in the terms are 4, 3, and 3. Choose the smallest power, which is 3.- The powers of \(n\) in the terms are 1, 2, and 3. Choose the smallest power, which is 1.

Step 5: Combine the GCF of Coefficients and Variables

Combine the results from Step 3 and Step 4 to form the greatest common factor (GCF). This means multiplying the GCF of the coefficients, which is 3, with the smallest powers of both variables, which are \(m^3\) and \(n^1\). Therefore, the GCF is \(3m^3n\).

Key Concepts

Coefficients

When finding the Greatest Common Factor (GCF) of a list of algebraic terms, it's important to address each component separately. One such component is the coefficients.
Coefficients are the numerical parts of the terms in an expression. In the given exercise, the terms are:

• 6
• 12
• 9

These are the coefficients that we need to analyze to find their GCF. To do this, we first list out all the factors of each coefficient separately. Factors are numbers that multiply together to give the original number.
For instance:

• The factors of 6 are 1, 2, 3, and 6.
• The factors of 12 include 1, 2, 3, 4, 6, and 12.
• 9 is factored into 1, 3, and 9.

The common factors among these numbers are identified as 1 and 3. Among these, 3 is the largest, making it the GCF of the coefficients. This step isolates the pure numerical part of the expression, simplifying further calculations.

Variables

Variables are the alphabetic portions of algebraic terms that represent unknown values and can vary. In our exercise, these variables are represented by the letters "m" and "n." Each term can have different powers of these variables:

• For example, we have variables in different powers across the three terms:
  • First term: 6 \( m^{4} n \)
  • Second term: 12 \( m^{3} n^{2} \)
  • Third term: 9 \( m^{3} n^{3} \)

To find the GCF concerning these variables, we need to choose the smallest power for each variable present in the terms.

• For "m," the smallest power is 3, since we have \( m^{4} \), \( m^{3} \), and \( m^{3} \).
• For "n," the smallest power is 1, from the terms \( n \), \( n^{2} \), and \( n^{3} \).

Combining these, \( m^{3} n^{1} \) becomes part of the overall GCF of the terms. Understanding variables is crucial because they dictate how functions work and can change based on different inputs.

Factors

Factors play a crucial role in simplifying and solving algebraic expressions. In finding the GCF, we look at both numerical and variable factors combined. The purpose is to find the largest expression that divides each term in the given list.

• Numerical Factors:These are derived from the coefficients. As previously explained, the GCF of the coefficients is determined by finding common numerical factors. In our exercise, these were 1 and 3, with 3 being the greatest.
• Variable Factors:Each term's variable part must be analyzed by their powers. By choosing the smallest power for each variable across all terms, we maintain consistency and ensure the factor is valid for every term. This ensures that the GCF is not larger than any existing term.

Together, these factors (both numerical and variable) are multiplied to find the overall GCF. In this scenario, combining 3, \( m^3 \), and \( n^1 \), we arrive at a GCF of \( 3m^3n \). By understanding factors, one can break down complex expressions to their simplest forms, easing further mathematical manipulation.