Using the Greatest Common Factor and Factor by Grouping
Find the Greatest Common Factor of Two or More Expressions
Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.
We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
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.
First we'll find the GCF of two numbers.
Example 7.1
How to Find the Greatest Common Factor of Two or More Expressions
Find the GCF of 54 and 36.
Solution
| Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form. | Factor \(\underline{54}\) and \(\underline{36}\). |  |
| Step 2. In each column, circle the common factors. | Circle the 2,3, and 3 that are shared by both numbers. |  |
| Step 3. Bring down the common factors that all expressions share. | Bring down the 2,3, and 3 and then multiply. | \(GCF=2 \cdot \qquad 3 \cdot 3\) |
| Step 4. Multiply the factors. | \(GCF=18 \) The GCF of 54 and 36 is 18. |
Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.
\(54=18 \cdot 3\)
\(36=18 \cdot 2\)
Try It 7.1
Find the GCF of 48 and 80.
Try It 7.2
Find the GCF of 18 and 40.
We summarize the steps we use to find the GCF below.
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.
In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.
Example 7.2
Find the greatest common factor of \(27 x^{3}\) and \(18 x^{4}\).
Solution
| Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. |  |
| Bring down the common factors. |
\(\mathrm{GCF}=3 \cdot 3 \cdot \ qquad x \cdot x \cdot x\) |
| Multiply the factors. | \(\mathrm{GCF}=9 x^{3}\) |
| The GCF of \(27 x^{3}\) and \(18 x^{4}\) is \(9 x^{3}\). |
Try It 7.3
Find the GCF: \(12 x^{2}, 18 x^{3}\).
Try It 7.4
Find the GCF: \(16 y^{2}, 24 y^{3}\).
Example 7.3
Find the GCF of \(4 x^{2} y, 6 x y^{3}\).
Solution
| Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. |  |
| Bring down the common factors. | \(\mathrm{GCF}=2 \cdot \qquad x \cdot \quad y\) |
| Multiply the factors. | \(GCF=2xy\) |
| The GCF of \(4 x^{2} y\) and \(6 x y^{3}\) is \(2 \mathrm{xy}\). |
Try It 7.5
Find the GCF: \(6 a b^{4}, 8 a^{2} b\).
Try It 7.6
Find the GCF: \(9 m^{5} n^{2}, 12 m^{3} n\).
Example 7.4
Find the GCF of: \(21 x^{3}, 9 x^{2}, 15 x\).
Solution
| Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. |  |
| Bring down the common factors. | \(G C F=3 \cdot \qquad \quad x\) |
| Multiply the factors. | \(\mathrm{GCF}=3x\) |
| The GCF of \(21 x^{3}, 9 x^{2}\) and \(15 x\) is \(3 x\). |
Try It 7.7
Find the greatest common factor: \(25 m^{4}, 35 m^{3}, 20 m^{2}\).
Try It 7.8
Find the greatest common factor: \(14 x^{3}, 70 x^{2}, 105 x\).