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Study Guides > Prealgebra

Finding the Greatest Common Factor from Two Expressions

Learning Outcomes

  • Find the greatest common factor of two numbers
  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. On the left, the equation 8 times 7 equals 56 is shown. 8 and 7 are labeled factors, 56 is labeled product. On the right, the equation 2x times parentheses x plus 3 equals 2 x squared plus 6x is shown. 2x and x plus 3 are labeled factors, 2 x squared plus 6x is labeled product. There is an arrow on top pointing to the right that says In The Language of Algebra we factored 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 will find the greatest common factor of two numbers.

example

Find the greatest common factor of [latex]24[/latex] and [latex]36[/latex]. Solution
Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. Factor [latex]24[/latex] and [latex]36[/latex]. .
Step 2: List all factors--matching common factors in a column. .
In each column, circle the common factors. Circle the [latex]2, 2[/latex], and [latex]3[/latex] that are shared by both numbers. .
Step 3: Bring down the common factors that all expressions share. Bring down the [latex]2, 2, 3[/latex] and then multiply.
Step 4: Multiply the factors. The GCF of [latex]24[/latex] and [latex]36[/latex] is [latex]12[/latex].
Notice that since the GCF is a factor of both numbers, [latex]24[/latex] and [latex]36[/latex] can be written as multiples of [latex]12[/latex]. [latex-display]\begin{array}{c}24=12\cdot 2\\ 36=12\cdot 3\end{array}[/latex-display]
 

try it

[ohm_question]146326[/ohm_question]
  In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

Find the greatest common factor

  1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
  2. List all factors—matching common factors in a column. In each column, circle the common factors.
  3. Bring down the common factors that all expressions share.
  4. Multiply the factors.
 

example

Find the greatest common factor of [latex]5x\text{ and }15[/latex].

Answer: Solution

Factor each number into primes. Circle the common factors in each column. Bring down the common factors. .
The GCF of [latex]5x[/latex] and [latex]15[/latex] is [latex]5[/latex].

 

try it

[ohm_question]146327[/ohm_question]
  In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.  

example

Find the greatest common factor of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex].

Answer: 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. Multiply the factors. .
The GCF of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex] is [latex]6{x}^{2}[/latex]

 

try it

[ohm_question]146328[/ohm_question]
   

example

Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[/latex].

Answer: 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. Multiply the factors. .
The GCF of [latex]14{x}^{3}[/latex] and [latex]8{x}^{2}[/latex] and [latex]10x[/latex] is [latex]2x[/latex]

 

try it

[ohm_question]146329[/ohm_question]
Watch the following video to see another example of how to find the GCF of two monomials that have one variable. https://youtu.be/EhkVBXRBC2s

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  • Question ID 146329, 146328, 146327, 146326. Authored by: Lumen Learning. License: CC BY: Attribution.

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  • Ex: Determine the GCF of Two Monomials (One Variables). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.

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