Greatest Common Factor
The greatest common factor (GCF) of two or more expressions is the largest factor that divides all of them evenly. To find the GCF of 12x squared and 18x, identify the GCF of the coefficients (6) and the GCF of the variable parts (x), giving a GCF of 6x. In Openstax Elementary Algebra 2E, students use the GCF as the first step in factoring polynomials, since factoring out the GCF simplifies the remaining expression. Finding the GCF is a prerequisite for all factoring techniques in algebra.
Key Concepts
Property The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
HOW TO: Find the Greatest Common Factor (GCF) of two expressions. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors. Step 3. Bring down the common factors that all expressions share. Step 4. Multiply the factors.
Examples Find the GCF of $42$ and $70$. We factor each number: $42 = 2 \cdot 3 \cdot 7$ and $70 = 2 \cdot 5 \cdot 7$. The common factors are $2$ and $7$. So, the GCF is $2 \cdot 7 = 14$. Find the GCF of $15a^2$ and $25a^3$. We factor each term: $15a^2 = 3 \cdot 5 \cdot a \cdot a$ and $25a^3 = 5 \cdot 5 \cdot a \cdot a \cdot a$. The common factors are $5, a, a$. The GCF is $5a^2$. Find the GCF of $12x^2y$ and $18xy^2$. We factor each term: $12x^2y = 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y$ and $18xy^2 = 2 \cdot 3 \cdot 3 \cdot x \cdot y \cdot y$. The common factors are $2, 3, x, y$. The GCF is $6xy$.
Common Questions
What is the greatest common factor (GCF)?
The GCF of two or more expressions is the largest expression that divides all of them evenly. For numbers, it is the largest integer that is a factor of all given numbers. For polynomials, it includes both the numeric GCF and the highest power of shared variables.
How do you find the GCF of two numbers?
List the factors of each number and identify the largest one they share. For 24 and 36, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 and the factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36. The GCF is 12.
How do you find the GCF of polynomial terms?
Find the GCF of the coefficients separately, then find the lowest power of each shared variable. For 8x cubed and 12x squared, the GCF of 8 and 12 is 4, and the lowest power of x is x squared, so the GCF is 4x squared.
Why do you find the GCF when factoring polynomials?
Factoring out the GCF first simplifies the remaining polynomial and makes subsequent factoring steps easier. It also ensures the factored form is fully simplified. Always check for a GCF before using other factoring methods.
What is the difference between GCF and LCM?
The GCF is the largest common factor of two numbers while the LCM (least common multiple) is the smallest number both divide into evenly. For 4 and 6, the GCF is 2 and the LCM is 12.