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Elementary Algebra 2e

7.1 Greatest Common Factor and Factor by Grouping

Elementary Algebra 2e7.1 Greatest Common Factor and Factor by Grouping
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Learning Objectives

By the end of this section, you will be able to:

  • Find the greatest common factor of two or more expressions
  • Factor the greatest common factor from a polynomial
  • Factor by grouping
Be Prepared 7.1

Before you get started, take this readiness quiz.

Factor 56 into primes.
If you missed this problem, review Example 1.7.

Be Prepared 7.2

Find the least common multiple of 18 and 24.
If you missed this problem, review Example 1.10.

Be Prepared 7.3

Simplify −3(6a+11)−3(6a+11).
If you missed this problem, review Example 1.135.

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get 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.

This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.

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.

First we’ll find the GCF of two numbers.

Example 7.1

How to Find the Greatest Common Factor of Two or More Expressions

Find the GCF of 54 and 36.

Try It 7.1

Find the GCF of 48 and 80.

Try It 7.2

Find the GCF of 18 and 40.

We summarize the steps we use to find the GCF below.

How To

Find the Greatest Common Factor (GCF) of two expressions.

  1. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
  2. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
  3. Step 3. Bring down the common factors that all expressions share.
  4. Step 4. Multiply the factors.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Example 7.2

Find the greatest common factor of 27x3and18x427x3and18x4.

Try It 7.3

Find the GCF: 12x2,18x312x2,18x3.

Try It 7.4

Find the GCF: 16y2,24y316y2,24y3.

Example 7.3

Find the GCF of 4x2y,6xy34x2y,6xy3.

Try It 7.5

Find the GCF: 6ab4,8a2b6ab4,8a2b.

Try It 7.6

Find the GCF: 9m5n2,12m3n9m5n2,12m3n.

Example 7.4

Find the GCF of: 21x3,9x2,15x21x3,9x2,15x.

Try It 7.7

Find the greatest common factor: 25m4,35m3,20m225m4,35m3,20m2.

Try It 7.8

Find the greatest common factor: 14x3,70x2,105x14x3,70x2,105x.

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as 2·6or3·4),2·6or3·4), in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

2(x+7)factors2·x+2·72x+14product2(x+7)factors2·x+2·72x+14product

Now we will start with a product, like 2x+142x+14, and end with its factors, 2(x+7)2(x+7). 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,ca,b,c are real numbers, then

a(b+c)=ab+acandab+ac=a(b+c)a(b+c)=ab+acandab+ac=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 7.5

How to Factor the Greatest Common Factor from a Polynomial

Factor: 4x+124x+12.

Try It 7.9

Factor: 6a+246a+24.

Try It 7.10

Factor: 2b+142b+14.

How To

Factor the greatest common factor from a polynomial.

  1. Step 1. Find the GCF of all the terms of the polynomial.
  2. Step 2. Rewrite each term as a product using the GCF.
  3. Step 3. Use the “reverse” Distributive Property to factor the expression.
  4. Step 4. Check by multiplying the factors.

Factor as a Noun and a Verb

We use “factor” as both a noun and a verb.

This figure has two statements. The first statement has “noun”. Beside it the statement “7 is a factor of 14” labeling the word factor as the noun. The second statement has “verb”. Beside this statement is “factor 3 from 3a + 3 labeling factor as the verb.

Example 7.6

Factor: 5a+55a+5.

Try It 7.11

Factor: 14x+1414x+14.

Try It 7.12

Factor: 12p+1212p+12.

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example 7.7

Factor: 12x6012x60.

Try It 7.13

Factor: 18u3618u36.

Try It 7.14

Factor: 30y6030y60.

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Example 7.8

Factor: 4y2+24y+284y2+24y+28.

Try It 7.15

Factor: 5x225x+155x225x+15.

Try It 7.16

Factor: 3y212y+273y212y+27.

Example 7.9

Factor: 5x325x25x325x2.

Try It 7.17

Factor: 2x3+12x22x3+12x2.

Try It 7.18

Factor: 6y315y26y315y2.

Example 7.10

Factor: 21x39x2+15x21x39x2+15x.

Try It 7.19

Factor: 20x310x2+14x20x310x2+14x.

Try It 7.20

Factor: 24y312y220y24y312y220y.

Example 7.11

Factor: 8m312m2n+20mn28m312m2n+20mn2.

Try It 7.21

Factor: 9xy2+6x2y2+21y39xy2+6x2y2+21y3.

Try It 7.22

Factor: 3p36p2q+9pq33p36p2q+9pq3.

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example 7.12

Factor: −8y24−8y24.

Try It 7.23

Factor: −16z64−16z64.

Try It 7.24

Factor: −9y27−9y27.

Example 7.13

Factor: −6a2+36a−6a2+36a.

Try It 7.25

Factor: −4b2+16b−4b2+16b.

Try It 7.26

Factor: −7a2+21a−7a2+21a.

Example 7.14

Factor: 5q(q+7)6(q+7)5q(q+7)6(q+7).

Try It 7.27

Factor: 4m(m+3)7(m+3)4m(m+3)7(m+3).

Try It 7.28

Factor: 8n(n4)+5(n4)8n(n4)+5(n4).

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating 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 7.15

How to Factor by Grouping

Factor: xy+3y+2x+6xy+3y+2x+6.

Try It 7.29

Factor: xy+8y+3x+24xy+8y+3x+24.

Try It 7.30

Factor: ab+7b+8a+56ab+7b+8a+56.

How To

Factor by grouping.

  1. Step 1. Group terms with common factors.
  2. Step 2. Factor out the common factor in each group.
  3. Step 3. Factor the common factor from the expression.
  4. Step 4. Check by multiplying the factors.

Example 7.16

Factor: x2+3x2x6x2+3x2x6.

Try It 7.31

Factor: x2+2x5x10x2+2x5x10.

Try It 7.32

Factor: y2+4y7y28y2+4y7y28.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

Section 7.1 Exercises

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

1.

8, 18

2.

24, 40

3.

72, 162

4.

150, 275

5.

10a, 50

6.

5b, 30

7.

3x,10x23x,10x2

8.

21b2,14b21b2,14b

9.

8w2,24w38w2,24w3

10.

30x2,18x330x2,18x3

11.

10p3q,12pq210p3q,12pq2

12.

8a2b3,10ab28a2b3,10ab2

13.

12m2n3,30m5n312m2n3,30m5n3

14.

28x2y4,42x4y428x2y4,42x4y4

15.

10a3,12a2,14a10a3,12a2,14a

16.

20y3,28y2,40y20y3,28y2,40y

17.

35x3,10x4,5x535x3,10x4,5x5

18.

27p2,45p3,9p427p2,45p3,9p4

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

19.

4x+204x+20

20.

8y+168y+16

21.

6m+96m+9

22.

14p+3514p+35

23.

9q+99q+9

24.

7r+77r+7

25.

8m88m8

26.

4n44n4

27.

9n639n63

28.

45b1845b18

29.

3x2+6x93x2+6x9

30.

4y2+8y44y2+8y4

31.

8p2+4p+28p2+4p+2

32.

10q2+14q+2010q2+14q+20

33.

8y3+16y28y3+16y2

34.

12x310x12x310x

35.

5x315x2+20x5x315x2+20x

36.

8m240m+168m240m+16

37.

12xy2+18x2y230y312xy2+18x2y230y3

38.

21pq2+35p2q228q321pq2+35p2q228q3

39.

−2x4−2x4

40.

−3b+12−3b+12

41.

5x(x+1)+3(x+1)5x(x+1)+3(x+1)

42.

2x(x1)+9(x1)2x(x1)+9(x1)

43.

3b(b2)13(b2)3b(b2)13(b2)

44.

6m(m5)7(m5)6m(m5)7(m5)

Factor by Grouping

In the following exercises, factor by grouping.

45.

xy+2y+3x+6xy+2y+3x+6

46.

mn+4n+6m+24mn+4n+6m+24

47.

uv9u+2v18uv9u+2v18

48.

pq10p+8q80pq10p+8q80

49.

b2+5b4b20b2+5b4b20

50.

m2+6m12m72m2+6m12m72

51.

p2+4p9p36p2+4p9p36

52.

x2+5x3x15x2+5x3x15

Mixed Practice

In the following exercises, factor.

53.

−20x10−20x10

54.

5x3x2+x5x3x2+x

55.

3x37x2+6x143x37x2+6x14

56.

x3+x2x1x3+x2x1

57.

x2+xy+5x+5yx2+xy+5x+5y

58.

5x3+3x25x35x3+3x25x3

Everyday Math

59.

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression w26ww26w, where w=w= width. Factor the greatest common factor from the polynomial.

60.

Height of a baseball The height of a baseball t seconds after it is hit is given by the expression −16t2+80t+4−16t2+80t+4. Factor the greatest common factor from the polynomial.

Writing Exercises

61.

The greatest common factor of 36 and 60 is 12. Explain what this means.

62.

What is the GCF of y4,y5,andy10y4,y5,andy10? Write a general rule that tells you how to find the GCF of ya,yb,andycya,yb,andyc.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

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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