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Study Guides > College Algebra CoRequisite Course

Greatest Common Factor

Learning Outcomes

  • Find the greatest common factor of a list of expressions
  • Find the greatest common factor of a polynomial
Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, 22 and 1010 are factors of 2020, as are 4,5,1,204, 5, 1, 20. To factor a number is to rewrite it as a product. 20=4520=4\cdot{5} or 20=12020=1\cdot{20}. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – the action of rewriting a sum or difference as a product. Factoring is very helpful in simplifying expressions and solving equations involving polynomials. The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers: 44 is the GCF of 1616 and 2020 because it is the largest number that divides evenly into both 1616 and 2020. The GCF of two or more algebraic expressions works the same way: 4x4x is the GCF of 16x16x and 20x220{x}^{2} because it is the largest algebraic expression that divides evenly into both 16x16x and 20x220{x}^{2}.

Find the GCF of a list of algebraic expressions

We begin by finding the GCF of a list of numbers, then we'll extend the technique to monomial expressions containing variables. A good technique for finding the GCF of a list of numbers is to write each number as a product of its prime factors. Then, match all the common factors between each prime factorization. The product of all the common factors will build the greatest common factor.

example

Find the greatest common factor of 2424 and 3636.

Answer:

Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. Factor 2424 and 3636. .
Step 2: List all factors--matching common factors in a column. .
In each column, circle the common factors. Circle the 2,22, 2, and 33 that are shared by both numbers. .
Step 3: Bring down the common factors that all expressions share. Bring down the 2,2,32, 2, 3 and then multiply.
Step 4: Multiply the factors. The GCF of 2424 and 3636 is 1212.
Notice that since the GCF is a factor of both numbers, 2424 and 3636 can be written as multiples of 1212.

24=12236=123\begin{array}{c}24=12\cdot 2\\ 36=12\cdot 3\end{array}

Here's a summary of the technique.

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.

try it

[ohm_question]146326[/ohm_question]
In the previous example, we found the greatest common factor of a list of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. To find the GCF of an expression containing variable terms, first find the GCF of the coefficients, then find the GCF of the variables. The GCF of the variables will be the smallest degree of each variable that appears in each term. Here's an example using the matching method from the example above.

example

Find the greatest common factor of 5x and 155x\text{ and }15.

Answer: Solution

Factor each number into primes. Circle the common factors in each column. Bring down the common factors. .
The GCF of 5x5x and 1515 is 55.

 

try it

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

example

Find the greatest common factor of 12x212{x}^{2} and 18x318{x}^{3}.

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 12x212{x}^{2} and 18x318{x}^{3} is 6x26{x}^{2}

 

try it

[ohm_question]146328[/ohm_question]
Here are some examples of finding the GCF of a list of more than two expressions.

example

Find the greatest common factor of 14x3,8x2,10x14{x}^{3},8{x}^{2},10x.

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 14x314{x}^{3} and 8x28{x}^{2} and 10x10x is 2x2x

 

try it

[ohm_question]146329[/ohm_question]
The common-factor matching method works well for finding the GCF of the coefficients, but when finding the GCF of the variables, you may have noticed that we can simply select the smallest power of a variable that appears in each term. In fact, the GCF of a set of expressions in the form xn{x}^{n} will always be the exponent of lowest degree. 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 Sometimes you may encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example we find the GCF of two terms which both contain two variables.

Example

Find the greatest common factor of 81c3d81c^{3}d and 45c2d245c^{2}d^{2}.

Answer:

   81c3d=3333cccd45c2d2=335ccdd    GCF=33ccd\begin{array}{l}\,\,\,81c^{3}d=3\cdot3\cdot3\cdot3\cdot{c}\cdot{c}\cdot{c}\cdot{d}\\45c^{2}d^{2}=3\cdot3\cdot5\cdot{c}\cdot{c}\cdot{d}\cdot{d}\\\,\,\,\,\text{GCF}=3\cdot3\cdot{c}\cdot{c}\cdot{d}\end{array}

Answer

GCF=9c2d\text{GCF}=9c^{2}d

The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable. https://youtu.be/GfJvoIO3gKQ

Find the GCF of a polynomial

Now that you have practiced finding the GCF of a term with one and two variables, the next step is to find the GCF of a polynomial. Later in this module we will apply this idea to factoring the GCF out of a polynomial. That is, doing the distributive property "backwards" to divide the GCF away from each of the terms in the polynomial. In preparation, practice finding the GCF of a given polynomial. Recall that a polynomial is an expression consisting of a sum or difference of terms. To find the GCF of a polynomial, inspect each term for common factors just as you previously did with a list of expressions. No matter how large the polynomial, you can use the same technique described below to identify its GCF.

How To: Given a polynomial expression, find the greatest common factor.

  1. Identify the GCF of the coefficients.
  2. Identify the GCF of the variables.
  3. Combine to find the GCF of the expression.
 

Example

Find the GCF of 6x3y3+45x2y2+21xy6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy.

Answer: The GCF of 6,456,45, and 2121 is 33. The GCF of x3,x2{x}^{3},{x}^{2}, and xx is xx. And the GCF of y3,y2{y}^{3},{y}^{2}, and yy is yy. Combine these to find the GCF of the polynomial, 3xy3xy.

 

try it

[ohm_question]14137[/ohm_question]

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