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Intermediate Algebra 2e

6.1 Greatest Common Factor and Factor by Grouping

Intermediate Algebra 2e6.1 Greatest Common Factor and Factor by Grouping
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. 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 6.1

Before you get started, take this readiness quiz.

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: −3a(7a+8b).−3a(7a+8b).
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.

8 times 7 is 56. Here 8 and 7 are factors and 56 is the product. An arrow pointing from 8 times 7 to 56 is labeled multiply. An arrow pointing from 56 to 8 times 7 is labeled factor. 2x open parentheses x plus 3 close parentheses equals 2x squared plus 6x. Here the left side of the equation is labeled factors and the right side is labeled products.

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.

  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.

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 21x3,9x2,15x.21x3,9x2,15x.

Try It 6.1

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

Try It 6.2

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

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 2·62·6 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 3x2+15x,3x2+15x, and end with its factors, 3x(x+5).3x(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)=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 6.2

How to Use the Distributive Property to factor a polynomial

Factor: 8m312m2n+20mn2.8m312m2n+20mn2.

Try It 6.3

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

Try It 6.4

Factor: 3p36p2q+9pq3.3p36p2q+9pq3.

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:

Noun:7 is afactorof 14Verb:factor3 from3a+3Noun:7 is afactorof 14Verb:factor3 from3a+3

Example 6.3

Factor: 5x325x2.5x325x2.

Try It 6.5

Factor: 2x3+12x2.2x3+12x2.

Try It 6.6

Factor: 6y315y2.6y315y2.

Example 6.4

Factor: 8x3y10x2y2+12xy3.8x3y10x2y2+12xy3.

Try It 6.7

Factor: 15x3y3x2y2+6xy3.15x3y3x2y2+6xy3.

Try It 6.8

Factor: 8a3b+2a2b26ab3.8a3b+2a2b26ab3.

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

Example 6.5

Factor: −4a3+36a28a.−4a3+36a28a.

Try It 6.9

Factor: −4b3+16b28b.−4b3+16b28b.

Try It 6.10

Factor: −7a3+21a214a.−7a3+21a214a.

So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 6.6

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

Try It 6.11

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

Try It 6.12

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

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

Try It 6.13

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

Try It 6.14

Factor by grouping: ab+7b+8a+56.ab+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 6.8

Factor by grouping: x2+3x2x6x2+3x2x6 6x23x4x+2.6x23x4x+2.

Try It 6.15

Factor by grouping: x2+2x5x10x2+2x5x10 20x216x15x+12.20x216x15x+12.

Try It 6.16

Factor by grouping: y2+4y7y28y2+4y7y28 42m218m35m+15.42m218m35m+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.

1.

10p3q,12pq210p3q,12pq2

2.

8a2b3,10ab28a2b3,10ab2

3.

12m2n3,30m5n312m2n3,30m5n3

4.

28x2y4,42x4y428x2y4,42x4y4

5.

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

6.

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

7.

35x3y2,10x4y,5x5y335x3y2,10x4y,5x5y3

8.

27p2q3,45p3q4,9p4q327p2q3,45p3q4,9p4q3

Factor the Greatest Common Factor from a Polynomial

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

9.

6m+96m+9

10.

14p+3514p+35

11.

9n639n63

12.

45b1845b18

13.

3x2+6x93x2+6x9

14.

4y2+8y44y2+8y4

15.

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

16.

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

17.

8y3+16y28y3+16y2

18.

12x310x12x310x

19.

5x315x2+20x5x315x2+20x

20.

8m240m+168m240m+16

21.

24x312x2+15x24x312x2+15x

22.

24y318y230y24y318y230y

23.

12xy2+18x2y230y312xy2+18x2y230y3

24.

21pq2+35p2q228q321pq2+35p2q228q3

25.

20x3y4x2y2+12xy320x3y4x2y2+12xy3

26.

24a3b+6a2b218ab324a3b+6a2b218ab3

27.

−2x4−2x4

28.

−3b+12−3b+12

29.

−2x3+18x28x−2x3+18x28x

30.

−5y3+35y215y−5y3+35y215y

31.

−4p3q12p2q2+16pq2−4p3q12p2q2+16pq2

32.

−6a3b12a2b2+18ab2−6a3b12a2b2+18ab2

33.

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

34.

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

35.

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

36.

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

Factor by Grouping

In the following exercises, factor by grouping.

37.

ab+5a+3b+15ab+5a+3b+15

38.

cd+6c+4d+24cd+6c+4d+24

39.

8y2+y+40y+58y2+y+40y+5

40.

6y2+7y+24y+286y2+7y+24y+28

41.

uv9u+2v18uv9u+2v18

42.

pq10p+8q80pq10p+8q80

43.

u2u+6u6u2u+6u6

44.

x2x+4x4x2x+4x4

45.

9p2+12p15p209p2+12p15p20

46.

16q2+20q28q3516q2+20q28q35

47.

mn6m4n+24mn6m4n+24

48.

r23rr+3r23rr+3

49.

2x214x5x+352x214x5x+35

50.

4x236x3x+274x236x3x+27

Mixed Practice

In the following exercises, factor.

51.

−18xy227x2y−18xy227x2y

52.

−4x3y5x2y3+12xy4−4x3y5x2y3+12xy4

53.

3x37x2+6x143x37x2+6x14

54.

x3+x2+x+1x3+x2+x+1

55.

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

56.

5x33x2+5x35x33x2+5x3

Writing Exercises

57.

What does it mean to say a polynomial is in factored form?

58.

How do you check result after factoring a polynomial?

59.

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

60.

What is the GCF of y4,y5,y4,y5, and y10?y10? Write a general rule that tells you how to find the GCF of ya,yb,ya,yb, and yc.yc.

Self Check

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

This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no I don’t get it. The first column has the following statements: find the greatest common factor of 2 or more expressions, factor the greatest common factor from a polynomial, factor by grouping. The remaining columns are blank.

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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