Finding the Greatest Common Factor
Learning Outcomes
- Find the greatest common factor of multiple numbers
- Find the greatest common factor of monomials
But because the equation was written as the product of two terms, we could use the zero product principle. What if we are given a polynomial equation that is not written as a product of two terms, such as this one ? We can use a technique called factoring, where we try to find factors that can be divided into each term of the polynomial so it can be rewritten as a product.
In this section we will explore how to find common factors from the terms of a polynomial, and rewrite it as a product. This technique will help us solve polynomial equations in the next section.
Finding the Greatest Common Factor of Two Numbers
Earlier we multiplied factors together to get a product. Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: and are factors of , as are and and and . To factor a number is to rewrite it as a product. . In algebra, we use the word factor as both a noun - something being multiplied - and as a verb - rewriting a sum or difference as 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.
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.
Their greatest common factor is , since is the greatest factor that both numbers have in common. To find the GCF of greater numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together. A prime factor is similar to a prime number—it has only itself and 1 as factors. The process of breaking a number down into its prime factors is called prime factorization.
example
Find the greatest common factor of and . SolutionStep 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. | Factor and . | ![]() |
Step 2: List all factors--matching common factors in a column. | ![]() |
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In each column, circle the common factors. | Circle the , and that are shared by both numbers. | ![]() |
Step 3: Bring down the common factors that all expressions share. | Bring down the and then multiply. | |
Step 4: Multiply the factors. | The GCF of and is . |
Example
Find the greatest common factor of and .Answer:
Answer
Because the GCF is the product of the prime factors that these numbers have in common, you know that it is a factor of both numbers. (If you want to test this, go ahead and divide both and by —they are both evenly divisible by this number!)
The video that follows shows another example of finding the greatest common factor of two whole numbers.
https://youtu.be/KbBJcdDY_VE
try it
[ohm_question]146326[/ohm_question]Greatest Common Factor of Polynomials
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. To factor a polynomial, you rewrite it as a product. Any integer can be written as the product of factors, and we can apply this technique to monomials or polynomials. Factoring is very helpful in simplifying and solving equations using polynomials. Finding the greatest common factor in a set of monomials is not very different from finding the GCF of two whole numbers. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF. We summarize below a list of steps that can help you to find the greatest common factor.Find the greatest common factor
- Factor each coefficient into primes. Write all variables with exponents in expanded form.
- List all factors—matching common factors in a column. In each column, circle the common factors.
- Bring down the common factors that all expressions share.
- Multiply the factors.
example
Find the greatest common factor of .Answer: Solution
Factor each number into primes. Circle the common factors in each column. Bring down the common factors. | ![]() |
The GCF of and is . |
try it
[ohm_question]146327[/ohm_question]example
Find the greatest common factor of and .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 and is |
Example
Find the greatest common factor of and .Answer:
Answer
try it
[ohm_question]146328[/ohm_question]example
Find the greatest common factor of .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 and and is |
try it
[ohm_question]146329[/ohm_question]Example
Find the greatest common factor of and .Answer:
Answer
Try It
[ohm_question]39942[/ohm_question]Contribute!
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