Greatest Common Factor
Learning Objective(s)
· Find the greatest common factor (GCF) of monomials.
· Factor polynomials by factoring out the greatest common factor (GCF).
· Factor expressions with four terms by grouping.
Introduction
Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: 2 and 10 are factors of 20, as are 4 and 5 and 1 and 20. To factor a number is to rewrite it as a product. 20 = 4 • 5.
Likewise to factor a polynomial, you rewrite it as a product. Just as any integer can be written as the product of factors, so too can any monomial or polynomial be expressed as a product of factors. Factoring is very helpful in simplifying and solving equations using polynomials.
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.
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 50 and 30.
50 = 10 • 5
30 = 10 • 3
Their greatest common factor is 10, since 10 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.
Example  
Problem  Find the greatest common factor of 210 and 168. 
 210 = 2 • 3 • 5 • 7 
 168 = 2 • 2 • 2 • 3 • 7 
 GCF = 2 • 3 • 7 
Answer  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 210 and 168 by 42—they are both evenly divisible by this number!)
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.
Example  
Problem  Find the greatest common factor of 25b^{3} and 10b^{2}. 
 25b^{3} = 5 • 5 • b • b • b 
 10b^{2} = 5 • 2 • b • b 
 GCF = 5 • b • b 
Answer  GCF = 5b^{2} 
The monomials have the factors 5, b, and b in common, which means their greatest common factor is 5 • b • b, or simply 5b^{2}.
Example  
Problem  Find the greatest common factor of 81c^{3}d and 45c^{2}d^{2}. 
 81c^{3}d = 3 • 3 • 3 • 3 • c • c • c • d 
 45c^{2}d^{2} = 3 • 3 • 5 • c • c • d • d 
 GCF = 3 • 3 • c • c • d 
Answer  GCF = 9c^{2}d 
Find the greatest common factor of 56xy and 16y^{3}.
A) 8
B) 8y
C) 16y
D) 8xy^{3}

When two or more monomials are combined (either added or subtracted), the resulting expression is called a polynomial. If you can find common factors for each term of the polynomial, then you can factor the polynomial.
As you look at the examples of simple polynomials below, try to identify factors that the terms of the polynomial have in common.
Polynomial  Terms  Common Factors 
6x + 9  6x and 9  3 is a factor of 6x and 9 
a^{2} – 2a  a^{2} and −2a  a is a factor of a^{2} and −2a 
4c^{3} + 4c  4c^{3} and 4c  4 and c are factors of 4c^{3} and 4c 
To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the distributive property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.
Product of a number and a sum: a(b + c) = a • b + a • c. You can say that “a is being distributed over b + c.”
Sum of the products: a • b + a • c = a(b + c). Here you can say that “a is being factored out.”
In both cases, it is the distributive property that is being used.
 Example  
Problem 
Factor 25b^{3} + 10b^{2}. 
 
 25b^{3} = 5 • 5 • b • b • b 10b^{2} = 5 • 2 • b • b GCF = 5 • b • b = 5b^{2}  Find the GCF. From a previous example, you found the GCF of 25b^{3} and 10b^{2} to be 5b^{2}.  
 25b^{3} = 5b^{2} • 5b
10b^{2} = 5b^{2} • 2  Rewrite each term with the GCF as one factor.  

5b^{2}(5b) + 5b^{2}(2)  Rewrite the polynomial using the factored terms in place of the original terms.  
 5b^{2}(5b + 2)  Factor out the 5b^{2}.  
Answer  5b^{2}(5b + 2) 
 
The factored form of the polynomial 25b^{3} + 10b^{2} is 5b^{2}(5b + 2). You can check this by doing the multiplication. 5b^{2}(5b + 2) = 25b^{3} + 10b^{2}.
Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.
For example:
25b^{3} + 10b^{2 }= 5(5b^{3 }+ 2b^{2}) Factor out 5.
= 5b^{2}(5b + 2) Then factor out b^{2}.
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.
Example  
Problem  Factor 81c^{3}d + 45c^{2}d^{2}.  

3 • 3 • 9 • c • c • c • d  Factor 81c^{3}d. 
 3 • 3 • 5 • c • c • d • d  Factor 45c^{2}d^{2}. 
 3 • 3 • c • c • d = 9c^{2}d  Find the GCF. 
 81c^{3}d = 9c^{2}d(9c)
45c^{2}d^{2} = 9c^{2}d(5d)  Rewrite each term as the product of the GCF and the remaining terms. 

9c^{2}d(9c) + 9c^{2}d(5d) 
Rewrite the polynomial expression using the factored terms in place of the original terms. 

9c^{2}d(9c + 5d) 
Factor out 9c^{2}d. 
Answer  9c^{2}d(9c + 5d) 

Factor 8a^{6} – 11a^{5}.
A) 88(a^{6} – a^{5})
B) 8a(a^{5} – 3)
C) a^{5}(a – 1)
D) a^{5}(8a – 11)

The distributive property allows you to factor out common factors. However, what do you do if the terms within the polynomial do not share any common factors?
If there is no common factor for all of the terms in the polynomial, another technique needs to be used to see if the polynomial can be factored. It involves organizing the polynomial in groups.
Example  
Problem 
Factor 4ab + 12a + 3b + 9 

 (4ab + 12a) + (3b + 9)  Group terms into pairs. 
 4ab = 2 • 2 • a • b 12a = 3 • 2 • 2 • a
GCF = 4a  Find the GCF of the first pair of terms.

 (4a • b + 4a • 3) + (3b + 9) 4a(b + 3) + (3b + 9) 
Factor the GCF, 4a, out of the first group. 
 3b = 3 • b 9 = 3 • 3
GCF =3  Find the GCF of the second pair of terms.

 4a(b + 3) +(3 • b + 3 • 3) 4a(b + 3) + 3(b + 3) 
Factor 3 out of the second group. 
 4a(b + 3) + 3(b + 3)
(b + 3)(4a + 3)  Notice that the two terms have a common factor (b + 3).
Factor out the common factor (b + 3) from the two terms. 
Answer 
(b + 3)(4a + 3) 

Notice that when you factor two terms, the result is a monomial times a polynomial. But the factored form of a fourterm polynomial is the product of two binomials.
This process is called the grouping technique. Broken down into individual steps, here's how to do it (you can also follow this process in the example below).
Let’s try factoring a few more fourterm polynomials. Notice that in the example below, the first term is x^{2}, and x is the only variable present.
Example  
Problem 
Factor x^{2} + 2x + 5x + 10 

 (x^{2} + 2x) + (5x + 10)  Group terms of the polynomial into pairs. 

x(x + 2) + (5x + 10) 
Factor out the like factor, x, from the first group. 

x(x + 2) + 5(x + 2) 
Factor out the like factor, 5, from the second group. 

(x + 2)(x + 5) 
Look for common factors between the factored forms of the paired terms. Here, the common factor is (x + 2).
Factor out the common factor, (x + 2), from both terms. The polynomial is now factored. 
Answer  (x + 2)(x + 5) 

This method of factoring only works in some cases. Notice that both factors here contain the term x.
Example  
Problem 
Factor 2x^{2} – 3x + 8x – 12. 

 (2x^{2} – 3x) + (8x – 12)  Group terms into pairs. 

x(2x – 3) + 4(2x – 3) 
Factor the common factor, x, out of the first group and the common factor, 4, out of the second group. 

(x + 4)(2x – 3) 
Factor out the common factor, (2x – 3), from both terms. 
Answer  (x + 4)(2x – 3) 

Example  
Problem 
Factor 3x^{2} + 3x – 2x – 2. 

 (3x^{2} + 3x) + (−2x – 2)  Group terms into pairs. Since subtraction is the same as addition of the opposite, you can write −2x – 2 as + (−2x – 2). 

3x(x + 1) + (−2x – 2) 
Factor the common factor 3x out of first group. 

3x(x + 1) −2(x + 1) 
Factor the common factor −2 out of the second group. Notice what happens to the signs within the parentheses once −2 is factored out. 

(x + 1)(3x – 2) 
Factor out the common factor, (x + 1), from both terms. 
Answer  (x + 1)(3x – 2) 

Factor 10ab + 5b + 8a + 4.
A) (2a + 1)(5b + 4)
B) (5b + 2a)(4 + 1)
C) 5(2ab + b + 8a + 4)
D) (4 + 2a)(5b + 1)

Sometimes, you will encounter polynomials that, despite your best efforts, cannot be factored into the product of two binomials.
Example  
Problem 
Factor 7x^{2} – 21x + 5x – 5. 

 (7x^{2} – 21x) + (5x – 5)  Group terms into pairs. 

7x(x – 3) + (5x – 5) 
Factor the common factor 7x out of the first group. 

7x(x – 3) + 5(x – 1) 
Factor the common factor 5 out of the second group. 

7x(x – 3) + 5(x – 1) 
The two groups 7x(x – 3) and 5(x – 1) do not have any common factors, so this polynomial cannot be factored any further. 
Answer  Cannot be factored 

In the example above, each pair can be factored, but then there is no common factor between the pairs!
Summary
A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in a • b + a • c form has been rewritten as a(b + c), where a is the GCF, the polynomial is in factored form.
When factoring a fourterm polynomial using grouping, find the common factor of pairs of terms rather than the whole polynomial. Use the distributive property to rewrite the grouped terms as the common factor times a binomial. Finally, pull any common binomials out of the factored groups. The fully factored polynomial will be the product of two binomials.