6.1 Greatest Common Factor and Factor by Grouping
Factor 56 into primes.
If you missed this problem, review Example 1.2.
Be Prepared 6.2
Find the least common multiple (LCM) of 18 and 24.
If you missed this problem, review Example 1.3.
Be Prepared 6.3
Multiply: −3 a ( 7 a + 8 b ) . −3 a ( 7 a + 8 b ) .
If you missed this problem, review Example 5.26.
Find the Greatest Common Factor of Two or More Expressions
Earlier we multiplied factors together to get a product . Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring .
We have learned how to factor 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.
We summarize the steps we use to find the greatest common factor.
How To
Find the greatest common factor (GCF) of two expressions.
- Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
- Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
- Step 3. Bring down the common factors that all expressions share.
- Step 4. Multiply the factors.
The next example will show us the steps to find the greatest common factor of three expressions.
Example 6.1
Find the greatest common factor of 21 x 3 , 9 x 2 , 15 x . 21 x 3 , 9 x 2 , 15 x .
Solution
15 x is written as 3 times 5 times x. Now bring down the common factors. So, GCF is 3 times x. Now multiply the factors. So, GCF of 21 x cubed, 9 x squared and 15x is 3x." data-label=""> 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 21 x 3 21 x 3 , 9 x 2 9 x 2 and 15 x 15 x is 3 x 3 x . |
Find the greatest common factor: 25 m 4 , 35 m 3 , 20 m 2 . 25 m 4 , 35 m 3 , 20 m 2 .
Find the greatest common factor: 14 x 3 , 70 x 2 , 105 x . 14 x 3 , 70 x 2 , 105 x .
Factor the Greatest Common Factor from a Polynomial
It is sometimes useful to represent a number as a product of factors, for example, 12 as 2 · 6 2 · 6 or 3 · 4 . 3 · 4 . In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as 3 x 2 + 15 x , 3 x 2 + 15 x , and end with its factors, 3 x ( x + 5 ) . 3 x ( x + 5 ) . To do this we apply the Distributive Property “in reverse.”
We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”
Distributive Property
If a, b, and c are real numbers, then
a ( b + c ) = a b + a c and a b + a c = a ( b + c ) a ( b + c ) = a b + a c and a b + a c = a ( b + c )
The form on the left is used to multiply. The form on the right is used to factor.
So how do you use the Distributive Property to factor a polynomial ? You just find the GCF of all the terms and write the polynomial as a product!
Example 6.2
How to Use the Distributive Property to factor a polynomial
Factor: 8 m 3 − 12 m 2 n + 20 m n 2 . 8 m 3 − 12 m 2 n + 20 m n 2 .
Solution
Factor: 9 x y 2 + 6 x 2 y 2 + 21 y 3 . 9 x y 2 + 6 x 2 y 2 + 21 y 3 .
Factor: 3 p 3 − 6 p 2 q + 9 p q 3 . 3 p 3 − 6 p 2 q + 9 p q 3 .
How To
Factor the greatest common factor from a polynomial.
- Step 1. Find the GCF of all the terms of the polynomial.
- Step 2. Rewrite each term as a product using the GCF.
- Step 3. Use the “reverse” Distributive Property to factor the expression.
- Step 4. Check by multiplying the factors.
Factor as a Noun and a Verb
We use “factor” as both a noun and a verb:
Noun: 7 is a factor of 14 Verb: factor 3 from 3 a + 3 Noun: 7 is a factor of 14 Verb: factor 3 from 3 a + 3
Example 6.3
Factor: 5 x 3 − 25 x 2 . 5 x 3 − 25 x 2 .
Solution
polynomial as 5 x squared times x minus 5 x squared times 5. Next we factor the GCF and rewrite as 5 x squared open parentheses x minus 5 close parentheses. Finally we check by multiplying the factors to get the original polynomial." data-label=""> | Find the GCF of 5 x 3 5 x 3 and 25 x 2 . 25 x 2 . |
| Rewrite each term. |
| Factor the GCF. |
Check:
5 x 2 ( x − 5 ) 5 x 2 · x − 5 x 2 · 5 5 x 3 − 25 x 2 ✓ 5 x 2 ( x − 5 ) 5 x 2 · x − 5 x 2 · 5 5 x 3 − 25 x 2 ✓ |
Factor: 2 x 3 + 12 x 2 . 2 x 3 + 12 x 2 .
Factor: 6 y 3 − 15 y 2 . 6 y 3 − 15 y 2 .
Example 6.4
Factor: 8 x 3 y − 10 x 2 y 2 + 12 x y 3 . 8 x 3 y − 10 x 2 y 2 + 12 x y 3 .
Solution
The GCF of 8 x 3 y , −10 x 2 y 2 , and 12 x y 3 8 x 3 y , −10 x 2 y 2 , and 12 x y 3
is 2 x y . 2 x y . |
| Rewrite each term using the GCF, 2 x y . 2 x y . |
| Factor the GCF. |
Check:
2 x y ( 4 x 2 − 5 x y + 6 y 2 ) 2 x y · 4 x 2 − 2 x y · 5 x y + 2 x y · 6 y 2 8 x 3 y − 10 x 2 y 2 + 12 x y 3 ✓ 2 x y ( 4 x 2 − 5 x y + 6 y 2 ) 2 x y · 4 x 2 − 2 x y · 5 x y + 2 x y · 6 y 2 8 x 3 y − 10 x 2 y 2 + 12 x y 3 ✓ |
Factor: 15 x 3 y − 3 x 2 y 2 + 6 x y 3 . 15 x 3 y − 3 x 2 y 2 + 6 x y 3 .
Factor: 8 a 3 b + 2 a 2 b 2 − 6 a b 3 . 8 a 3 b + 2 a 2 b 2 − 6 a b 3 .
When the leading coefficient is negative, we factor the negative out as part of the GCF.
Example 6.5
Factor: −4 a 3 + 36 a 2 − 8 a . −4 a 3 + 36 a 2 − 8 a .
Solution
The leading coefficient is negative, so the GCF will be negative.
| Rewrite each term using the GCF, −4 a . −4 a . |
| Factor the GCF. |
Check:
−4 a ( a 2 − 9 a + 2 ) −4 a · a 2 − ( −4 a ) · 9 a + ( −4 a ) · 2 −4 a 3 + 36 a 2 − 8 a ✓ −4 a ( a 2 − 9 a + 2 ) −4 a · a 2 − ( −4 a ) · 9 a + ( −4 a ) · 2 −4 a 3 + 36 a 2 − 8 a ✓ |
Factor: −4 b 3 + 16 b 2 − 8 b . −4 b 3 + 16 b 2 − 8 b .
Factor: −7 a 3 + 21 a 2 − 14 a . −7 a 3 + 21 a 2 − 14 a .
So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.
Example 6.6
Factor: 3 y ( y + 7 ) − 4 ( y + 7 ) . 3 y ( y + 7 ) − 4 ( y + 7 ) .
Solution
The GCF is the binomial y + 7 . y + 7 .
| Factor the GCF, ( y + 7 ) . ( y + 7 ) . |
| Check on your own by multiplying. |
Factor: 4 m ( m + 3 ) − 7 ( m + 3 ) . 4 m ( m + 3 ) − 7 ( m + 3 ) .
Factor: 8 n ( n − 4 ) + 5 ( n − 4 ) . 8 n ( n − 4 ) + 5 ( n − 4 ) .
Factor by Grouping
Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime , some polynomials are prime.
Example 6.7
How to Factor a Polynomial by Grouping
Factor by grouping: x y + 3 y + 2 x + 6 . x y + 3 y + 2 x + 6 .
Solution
Factor by grouping: x y + 8 y + 3 x + 24 . x y + 8 y + 3 x + 24 .
Factor by grouping: a b + 7 b + 8 a + 56 . a b + 7 b + 8 a + 56 .
How To
Factor by grouping.
- Step 1. Group terms with common factors.
- Step 2. Factor out the common factor in each group.
- Step 3. Factor the common factor from the expression.
- Step 4. Check by multiplying the factors.
Example 6.8
Factor by grouping: ⓐ x 2 + 3 x − 2 x − 6 x 2 + 3 x − 2 x − 6 ⓑ 6 x 2 − 3 x − 4 x + 2 . 6 x 2 − 3 x − 4 x + 2 .
Solution
| There is no GCF in all four terms. | x 2 + 3 x − 2 x − 6 x 2 + 3 x − 2 x − 6 |
| Separate into two parts. | x 2 + 3 x −2 x − 6 x 2 + 3 x −2 x − 6 |
| Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms. | x ( x + 3 ) − 2 ( x + 3 ) x ( x + 3 ) − 2 ( x + 3 ) |
| Factor out the common factor. | ( x + 3 ) ( x − 2 ) ( x + 3 ) ( x − 2 ) |
| Check on your own by multiplying. |
| There is no GCF in all four terms. | 6 x 2 − 3 x − 4 x + 2 6 x 2 − 3 x − 4 x + 2 |
| Separate into two parts. | 6 x 2 − 3 x −4 x + 2 6 x 2 − 3 x −4 x + 2 |
| Factor the GCF from both parts. | 3 x ( 2 x − 1 ) − 2 ( 2 x − 1 ) 3 x ( 2 x − 1 ) − 2 ( 2 x − 1 ) |
| Factor out the common factor. | ( 2 x − 1 ) ( 3 x − 2 ) ( 2 x − 1 ) ( 3 x − 2 ) |
| Check on your own by multiplying. |
Factor by grouping: ⓐ x 2 + 2 x − 5 x − 10 x 2 + 2 x − 5 x − 10 ⓑ 20 x 2 − 16 x − 15 x + 12 . 20 x 2 − 16 x − 15 x + 12 .
Factor by grouping: ⓐ y 2 + 4 y − 7 y − 28 y 2 + 4 y − 7 y − 28 ⓑ 42 m 2 − 18 m − 35 m + 15 . 42 m 2 − 18 m − 35 m + 15 .
Section 6.1 Exercises
Practice Makes Perfect
Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
10 p 3 q , 12 p q 2 10 p 3 q , 12 p q 2
8 a 2 b 3 , 10 a b 2 8 a 2 b 3 , 10 a b 2
12 m 2 n 3 , 30 m 5 n 3 12 m 2 n 3 , 30 m 5 n 3
28 x 2 y 4 , 42 x 4 y 4 28 x 2 y 4 , 42 x 4 y 4
10 a 3 , 12 a 2 , 14 a 10 a 3 , 12 a 2 , 14 a
20 y 3 , 28 y 2 , 40 y 20 y 3 , 28 y 2 , 40 y
35 x 3 y 2 , 10 x 4 y , 5 x 5 y 3 35 x 3 y 2 , 10 x 4 y , 5 x 5 y 3
27 p 2 q 3 , 45 p 3 q 4 , 9 p 4 q 3 27 p 2 q 3 , 45 p 3 q 4 , 9 p 4 q 3
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
14 p + 35 14 p + 35
9 n − 63 9 n − 63
45 b − 18 45 b − 18
3 x 2 + 6 x − 9 3 x 2 + 6 x − 9
4 y 2 + 8 y − 4 4 y 2 + 8 y − 4
8 p 2 + 4 p + 2 8 p 2 + 4 p + 2
10 q 2 + 14 q + 20 10 q 2 + 14 q + 20
8 y 3 + 16 y 2 8 y 3 + 16 y 2
12 x 3 − 10 x 12 x 3 − 10 x
5 x 3 − 15 x 2 + 20 x 5 x 3 − 15 x 2 + 20 x
8 m 2 − 40 m + 16 8 m 2 − 40 m + 16
24 x 3 − 12 x 2 + 15 x 24 x 3 − 12 x 2 + 15 x
24 y 3 − 18 y 2 − 30 y 24 y 3 − 18 y 2 − 30 y
12 x y 2 + 18 x 2 y 2 − 30 y 3 12 x y 2 + 18 x 2 y 2 − 30 y 3
21 p q 2 + 35 p 2 q 2 − 28 q 3 21 p q 2 + 35 p 2 q 2 − 28 q 3
20 x 3 y − 4 x 2 y 2 + 12 x y 3 20 x 3 y − 4 x 2 y 2 + 12 x y 3
24 a 3 b + 6 a 2 b 2 − 18 a b 3 24 a 3 b + 6 a 2 b 2 − 18 a b 3
−3 b + 12 −3 b + 12
−2 x 3 + 18 x 2 − 8 x −2 x 3 + 18 x 2 − 8 x
−5 y 3 + 35 y 2 − 15 y −5 y 3 + 35 y 2 − 15 y
−4 p 3 q − 12 p 2 q 2 + 16 p q 2 −4 p 3 q − 12 p 2 q 2 + 16 p q 2
−6 a 3 b − 12 a 2 b 2 + 18 a b 2 −6 a 3 b − 12 a 2 b 2 + 18 a b 2
5 x ( x + 1 ) + 3 ( x + 1 ) 5 x ( x + 1 ) + 3 ( x + 1 )
2 x ( x − 1 ) + 9 ( x − 1 ) 2 x ( x − 1 ) + 9 ( x − 1 )
3 b ( b − 2 ) − 13 ( b − 2 ) 3 b ( b − 2 ) − 13 ( b − 2 )
6 m ( m − 5 ) − 7 ( m − 5 ) 6 m ( m − 5 ) − 7 ( m − 5 )
Factor by Grouping
In the following exercises, factor by grouping.
a b + 5 a + 3 b + 15 a b + 5 a + 3 b + 15
c d + 6 c + 4 d + 24 c d + 6 c + 4 d + 24
8 y 2 + y + 40 y + 5 8 y 2 + y + 40 y + 5
6 y 2 + 7 y + 24 y + 28 6 y 2 + 7 y + 24 y + 28
u v − 9 u + 2 v − 18 u v − 9 u + 2 v − 18
p q − 10 p + 8 q − 80 p q − 10 p + 8 q − 80
u 2 − u + 6 u − 6 u 2 − u + 6 u − 6
x 2 − x + 4 x − 4 x 2 − x + 4 x − 4
9 p 2 + 12 p − 15 p − 20 9 p 2 + 12 p − 15 p − 20
16 q 2 + 20 q − 28 q − 35 16 q 2 + 20 q − 28 q − 35
m n − 6 m − 4 n + 24 m n − 6 m − 4 n + 24
r 2 − 3 r − r + 3 r 2 − 3 r − r + 3
2 x 2 − 14 x − 5 x + 35 2 x 2 − 14 x − 5 x + 35
4 x 2 − 36 x − 3 x + 27 4 x 2 − 36 x − 3 x + 27
Mixed Practice
In the following exercises, factor.
−18 x y 2 − 27 x 2 y −18 x y 2 − 27 x 2 y
−4 x 3 y 5 − x 2 y 3 + 12 x y 4 −4 x 3 y 5 − x 2 y 3 + 12 x y 4
3 x 3 − 7 x 2 + 6 x − 14 3 x 3 − 7 x 2 + 6 x − 14
x 3 + x 2 + x + 1 x 3 + x 2 + x + 1
x 2 + x y + 5 x + 5 y x 2 + x y + 5 x + 5 y
5 x 3 − 3 x 2 + 5 x − 3 5 x 3 − 3 x 2 + 5 x − 3
Writing Exercises
What does it mean to say a polynomial is in factored form?
How do you check result after factoring a polynomial?
The greatest common factor of 36 and 60 is 12. Explain what this means.
What is the GCF of y 4 , y 5 , y 4 , y 5 , and y 10 ? y 10 ? Write a general rule that tells you how to find the GCF of y a , y b , y a , y b , and y c . y c .
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.
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