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Study Guides > ALGEBRA / TRIG I

Finding the Greatest Common Factor

Learning Outcomes

  • Find the greatest common factor of multiple numbers
  • Find the greatest common factor of monomials
In the section on the zero product principle, we showed that using the techniques for solving equations that we learned for linear equations did not work to solve

t(5t)=0t\left(5-t\right)=0

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 2y2+4y=02y^2+4y=0? 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: 22 and 1010 are factors of 2020, as are  44 and 55 and 11 and 2020. To factor a number is to rewrite it as a product. 20=4520=4\cdot5. 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. 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 We also 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. When we studied fractions, we learned that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 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 polynomials works the same way: 4x4x is the GCF of 16x16x and 20x220x^2 because it is the largest polynomial that divides evenly into both 16x16x and 20x220x^2.

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 show how to find the greatest common factor of two numbers.
Prime numbers written with dice including 5, 41, 19, 61, and many others Prime Numbers
  To get acquainted with the idea of factoring, let’s first find the greatest common factor (GCF) of two whole numbers. The GCF of two numbers is the greatest number that is a factor of both of the numbers. Take the numbers 5050 and 3030.

50=10530=103\begin{array}{l}50=10\cdot5\\30=10\cdot3\end{array}

Their greatest common factor is 1010, since 1010 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 2424 and 3636. Solution
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}

Example

Find the greatest common factor of 210210 and 168168.

Answer:

    210=2357    168=22237GCF=237\begin{array}{l}\,\,\,\,210=2\cdot3\cdot5\cdot7\\\,\,\,\,168=2\cdot2\cdot2\cdot3\cdot7\\\text{GCF}=2\cdot3\cdot7\end{array}

Answer

GCF=42\text{GCF}=42

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 210210 and168168 by 4242—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

  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 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 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 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}

 

Example

Find the greatest common factor of 25b325b^{3} and 10b210b^{2}.

Answer:

  25b3=55bbb  10b2=52bbGCF=5bb\begin{array}{l}\,\,25b^{3}=5\cdot5\cdot{b}\cdot{b}\cdot{b}\\\,\,10b^{2}=5\cdot2\cdot{b}\cdot{b}\\\text{GCF}=5\cdot{b}\cdot{b}\end{array}

Answer

GCF=5b2\text{GCF}=5b^{2}

The monomials have the factors 55, b, and b in common, which means their greatest common factor is 5bb5\cdot{b}\cdot{b}, or simply 5b25b^{2}.

try it

[ohm_question]146328[/ohm_question]
 

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]
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 We can also factor expressions that have more than one variable.

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

Try It

[ohm_question]39942[/ohm_question]
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

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