Greatest Common Factor (GCF) of Monomials

Step-by-Step Tutorial, Solved Examples, and Detailed Solutions

In algebra, the Greatest Common Factor (GCF) of two or more monomials is the largest monomial that divides evenly into all of them. This is a crucial skill for factoring polynomials and simplifying rational expressions.

The Prime Factorization Method

To find the GCF, we use prime factorization to break down each monomial into its building blocks (coefficients and variables).

  1. Write the prime factorization of each numerical coefficient.
  2. List the variables with their lowest appearing exponents in the set.
  3. Multiply the common numerical factors and variable factors together.

Worked Examples

Example 1: Two Monomials

Find the GCF of \( 12x \) and \( 18x^2 \).

View Solution

1. Prime factorization of \( 12x \):
\[ 12x = \color{red}{2} \cdot 2 \cdot \color{red}{3} \cdot \color{red}{x} \]

2. Prime factorization of \( 18x^2 \):
\[ 18x^2 = \color{red}{2} \cdot \color{red}{3} \cdot 3 \cdot \color{red}{x} \cdot x \]

3. Identify common factors: \( 2, 3, \text{ and } x \).

\[ \text{GCF}(12x, 18x^2) = \color{red}{2 \cdot 3 \cdot x = 6x} \]

Example 2: Multiple Variables

Find the GCF of \( 30x^2y^3 \), \( 42x^3y^2 \), and \( 18x^2y^2 \).

View Solution

Prime factorizations:

  • \( 30x^2y^3 = \color{red}{2 \cdot 3} \cdot 5 \cdot \color{red}{x \cdot x} \cdot \color{red}{y \cdot y} \cdot y \)
  • \( 42x^3y^2 = \color{red}{2 \cdot 3} \cdot 7 \cdot \color{red}{x \cdot x} \cdot x \cdot \color{red}{y \cdot y} \)
  • \( 18x^2y^2 = \color{red}{2 \cdot 3} \cdot 3 \cdot \color{red}{x \cdot x} \cdot \color{red}{y \cdot y} \)

The GCF is the product of common factors:

\[ \text{GCF} = 2 \cdot 3 \cdot x^2 \cdot y^2 = \color{red}{6x^2y^2} \]

Practice Questions with Detailed Solutions

Apply the prime factorization method to solve the following problems.

Question 1: Find the GCF of \( 36x^2 \) and \( 42x^3 \).

View Solution
\[ 36x^2 = \color{red}{2} \cdot 2 \cdot \color{red}{3} \cdot 3 \cdot \color{red}{x \cdot x} \] \[ 42x^3 = \color{red}{2 \cdot 3} \cdot 7 \cdot \color{red}{x \cdot x} \cdot x \]

GCF: \( 2 \cdot 3 \cdot x \cdot x = \color{red}{6x^2} \)

Question 2: Find the GCF of \( 45x^3 \), \( 60x^2 \), and \( 75x^4 \).

View Solution
\[ 45x^3 = \color{red}{3} \cdot 3 \cdot \color{red}{5} \cdot \color{red}{x \cdot x} \cdot x \] \[ 60x^2 = \color{red} 2 \cdot 2 \cdot {3} \cdot \color{red}{5} \cdot \color{red}{x \cdot x} \] \[ 75x^4 = \color{red}{3 \cdot 5} \cdot 5 \cdot \color{red}{x \cdot x} \cdot x \cdot x \]

GCF: \( 3 \cdot 5 \cdot x^2 = \color{red}{15x^2} \)

Question 3: Find the GCF of \( 50x^2y^3 \), \( 75x^2y^2 \), and \( 125x^4y^3 \).

View Solution
\[ 50x^2y^3 = \color{red} 2 \cdot {5 \cdot 5} \cdot \color{red}{x \cdot x} \cdot \color{red}{y \cdot y} \cdot y \] \[ 75x^2y^2 = \color{red} 3 \cdot {5 \cdot 5} \cdot \color{red}{x \cdot x} \cdot \color{red}{y \cdot y} \] \[ 125x^4y^3 = \color{red} 5 \cdot {5 \cdot 5} \cdot \color{red}{x \cdot x} \cdot x \cdot x \cdot \color{red}{y \cdot y} \cdot y \]

GCF: \( 5 \cdot 5 \cdot x^2 \cdot y^2 = \color{red}{25x^2y^2} \)

Question 4: Simplify the rational expression using the GCF: \[ \dfrac{35x^3y^2}{42x^2y^3} \]

View Solution

First, find the prime factorization for both numerator and denominator:

\[ 35x^3y^2 = 5 \cdot \color{red}{7 \cdot x^2 \cdot y^2} \cdot x \] \[ 42x^2y^3 = 2 \cdot 3 \cdot \color{red}{7 \cdot x^2 \cdot y^2} \cdot y \]

Cancel the common GCF factors \( (7x^2y^2) \):

\[ \dfrac{5x}{6y} \]

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