Finding the GCF of Two or More Expressions

6.4.2 Finding the GCF of Two or More Expressions

6.4.2 • Finding the GCF of Two or More Expressions

Activity

For the first part of the activity, you will work with a partner. Your teacher will hand out two pieces of paper, one to each of you. It is important for the activity that you do not show your partner your paper.

1. When you receive your paper, look at your expression. Write down all of the factors in your expression.

For example, if your expression was $100 x^{3} y^{2}$ , you would write down $2 \cdot 2 \cdot 5 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y$ . Do not share these factors with your partner yet.

2. When you and your partner have finished finding the factors of your expression, you will try to determine the GCF of these two terms without actually knowing the other’s expression! Take turns with your partner sharing one factor and the quantity of that factor. For example, you could say, “My expression has two factors that are 5” or “My expression has three factors that are $x$ .” Write down the factors your partner has shared with you.

3. Once you and your partner have finished taking turns sharing factors, you should know all of the factors in your partner’s expression. Using this information, determine your partner’s expression and find the GCF of these two expressions. Once you have found the GCF, compare your answer with your partner’s. Did you have the same answer?

Write the GCF that you found.

Compare your answer:

Expression A: $120 m^{3} n^{2} p^{4}$

Factors of Expression A: $2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot m \cdot m \cdot m \cdot n \cdot n \cdot p \cdot p \cdot p \cdot p$

Expression B: $80 m^{2} n s^{4}$

Factors of Expression B: $2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot m \cdot m \cdot n \cdot s \cdot s \cdot s \cdot s$

GCF: $2 \cdot 2 \cdot 2 \cdot 5 \cdot m \cdot m \cdot n = 40 m^{2} n$

Now, let’s look at an example of finding the GCF of two expressions as a class:

4. $18 g^{4} h^{2}, 63 g^{3} h^{3}$

Compare your answer:

Step 1 - Factor each coefficient into primes. Write all variables with exponents in expanded form.

$18 g^{4} h^{2} = 3 \cdot 3 \cdot 2 \cdot g \cdot g \cdot g \cdot g \cdot h \cdot h$

$63 g^{3} h^{3} = 3 \cdot 3 \cdot 7 \cdot g \cdot g \cdot g \cdot h \cdot h \cdot h$

Step 2 - List all factors, matching common factors in a column. In each column, circle the common factors.

Mathematical equations showing the prime factorization of 18 g to the fifth g squared and 63 g cubed h cubed with like factors grouped by red ovals for clarity.

Step 3 - Bring down the common factors that all expressions share.

GCF = $3 \cdot 3 \cdot g \cdot g \cdot g \cdot h \cdot h$

Step 4 - Multiply the factors.

GCF = $9 g^{3} h^{2}$

Work individually on the next part of the activity.

Find the greatest common factor of each of the following sets of expressions.

5. $10 p^{3} q, 12 p q^{2}$

Compare your answer:

$2 p q$

6. $8 a^{2} b^{3}, 10 a b^{2}$

Compare your answer:

$2 a b^{2}$

7. $10 a^{3}, 12 a^{2}, 14 a$

Compare your answer:

$2 a$

8. $35 x^{3} y^{2}, 10 x^{4} y, 5 x^{5} y^{3}$

Compare your answer:

$5 x^{3} y$

Self Check

Find the greatest common factor of

25m^4, 35m^3, 20m^2

$10m$
$5m^2$
$5m$
$10m^2$

Additional Resources

Finding the GCF of Two or More Expressions

Earlier, we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

Diagram showing multiplication and factoring. On the left: 8 × 7 = 56 with factors labeled under 8 and 7, and product under 56. On the right: 2x × (x + 3) = 2x² + 6x with factors and product labels. Arrows labeled multiply and factor.

Now we will factor expressions and find the greatest common factor of two or more expressions.

GREATEST COMMON FACTOR

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

We will summarize the steps we use to find the greatest common factor.

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.

The following example will show the steps to find the greatest common factor of three expressions.

Example

Find the greatest common factor of $21 x^{3}, 9 x^{2}, 15 x$ .

Step 1 - Factor each coefficient into primes and write the variables with exponents in expanded form.

Step 2 - Circle the common factors in each column.

Step 3 - Bring down the common factors.

Factoring example: 21x³, 9x², and 15x are broken into prime factors, with common factors 3 and x circled in red. The greatest common factor (GCF) is shown as 3x at the bottom.

Step 4 - Multiply the factors.

GFC = $3 x$

The GCF of $21 x^{3}$ , $9 x^{2}$ , and $15 x$ is $3 x$ .

Try it

Finding the GCF of Two or More Expressions

Find the greatest common factor of $14 x^{3}, 105 x, 70 x^{2}$ .

Here is how to find the GCF of multiple expressions.

Three algebraic expressions are shown with their factorizations. The number 7 and the variable x are circled in each case, highlighting common factors: 14x³, 105x, and 70x².

The GCF of these expressions is $7 x$ .