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Prealgebra 2e

10.6 Introduction to Factoring Polynomials

Prealgebra 2e10.6 Introduction to Factoring Polynomials
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. 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
  17. 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
Be Prepared 10.15

Before you get started, take this readiness quiz.

Factor 5656 into primes.
If you missed this problem, review Example 2.48.

Be Prepared 10.16

Multiply: −3(6a+11).−3(6a+11).
If you missed this problem, review Example 7.25.

Be Prepared 10.17

Multiply: 4x2(x2+3x1).4x2(x2+3x1).
If you missed this problem, review Example 10.32.

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.

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 “multiply” in red. There is an arrow on the bottom pointing to the left that says “factor” in red.

In The Language of Algebra we 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.

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 find the greatest common factor of two numbers.

Example 10.80

Find the greatest common factor of 2424 and 36.36.

Try It 10.159

Find the greatest common factor: 54,36.54,36.

Try It 10.160

Find the greatest common factor: 48,80.48,80.

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. We summarize the steps we use to find the greatest common factor.

How To

Find the greatest common factor.

  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.

Example 10.81

Find the greatest common factor of 5xand15.5xand15.

Try It 10.161

Find the greatest common factor: 7y,14.7y,14.

Try It 10.162

Find the greatest common factor: 22,11m.22,11m.

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 10.82

Find the greatest common factor of 12x212x2 and 18x3.18x3.

Try It 10.163

Find the greatest common factor: 16x2,24x3.16x2,24x3.

Try It 10.164

Find the greatest common factor: 27y3,18y4.27y3,18y4.

Example 10.83

Find the greatest common factor of 14x3,8x2,10x.14x3,8x2,10x.

Try It 10.165

Find the greatest common factor: 21x3,9x2,15x.21x3,9x2,15x.

Try It 10.166

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

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, 1212 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 greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

2(x + 7)factors 2·x + 2·7 2x + 14product 2(x + 7)factors 2·x + 2·7 2x + 14product

Here, we will start with a product, like 2x+14,2x+14, and end with its factors, 2(x+7).2(x+7). To do this we apply the Distributive Property “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 we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

Example 10.84

Factor: 2x+14.2x+14.

Try It 10.167

Factor: 4x+12.4x+12.

Try It 10.168

Factor: 6a+24.6a+24.

Notice that in Example 10.84, we used the word factor as both a noun and a verb:

Noun7is a factor of14Verbfactor2from2x+14Noun7is a factor of14Verbfactor2from2x+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 Distributive Property ‘in reverse’ to factor the expression.
  4. Step 4. Check by multiplying the factors.

Example 10.85

Factor: 3a+3.3a+3.

Try It 10.169

Factor: 9a+9.9a+9.

Try It 10.170

Factor: 11x+11.11x+11.

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 10.86

Factor: 12x60.12x60.

Try It 10.171

Factor: 11x44.11x44.

Try It 10.172

Factor: 13y52.13y52.

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

Example 10.87

Factor: 3y2+6y+9.3y2+6y+9.

Try It 10.173

Factor: 4y2+8y+12.4y2+8y+12.

Try It 10.174

Factor: 6x2+42x12.6x2+42x12.

In the next example, we factor a variable from a binomial.

Example 10.88

Factor: 6x2+5x.6x2+5x.

Try It 10.175

Factor: 9x2+7x.9x2+7x.

Try It 10.176

Factor: 5a212a.5a212a.

When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

Example 10.89

Factor: 4x320x2.4x320x2.

Try It 10.177

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

Try It 10.178

Factor: 6y315y2.6y315y2.

Example 10.90

Factor: 21y2+35y.21y2+35y.

Try It 10.179

Factor: 18y2+63y.18y2+63y.

Try It 10.180

Factor: 32k2+56k.32k2+56k.

Example 10.91

Factor: 14x3+8x210x.14x3+8x210x.

Try It 10.181

Factor: 18y36y224y.18y36y224y.

Try It 10.182

Factor: 16x3+8x212x.16x3+8x212x.

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

Example 10.92

Factor: −9y27.−9y27.

Try It 10.183

Factor: −5y35.−5y35.

Try It 10.184

Factor: −16z56.−16z56.

Pay close attention to the signs of the terms in the next example.

Example 10.93

Factor: −4a2+16a.−4a2+16a.

Try It 10.185

Factor: −7a2+21a.−7a2+21a.

Try It 10.186

Factor: −6x2+x.−6x2+x.

Media Access Additional Online Resources

Section 10.6 Exercises

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

422.

40,5640,56

423.

45,7545,75

424.

72,16272,162

425.

150,275150,275

426.

3x,123x,12

427.

4y,284y,28

428.

10a,5010a,50

429.

5b,305b,30

430.

16y,24y216y,24y2

431.

9x,15x29x,15x2

432.

18m3,36m218m3,36m2

433.

12p4,48p312p4,48p3

434.

10x,25x2,15x310x,25x2,15x3

435.

18a,6a2,22a318a,6a2,22a3

436.

24u,6u2,30u324u,6u2,30u3

437.

40y,10y2,90y340y,10y2,90y3

438.

15a4,9a5,21a615a4,9a5,21a6

439.

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

440.

27y2,45y3,9y427y2,45y3,9y4

441.

14b2,35b3,63b414b2,35b3,63b4

Factor the Greatest Common Factor from a Polynomial

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

442.

2x+82x+8

443.

5y+155y+15

444.

3a243a24

445.

4b204b20

446.

9y99y9

447.

7x77x7

448.

5m2+20m+355m2+20m+35

449.

3n2+21n+123n2+21n+12

450.

8p2+32p+488p2+32p+48

451.

6q2+30q+426q2+30q+42

452.

8q2+15q8q2+15q

453.

9c2+22c9c2+22c

454.

13k2+5k13k2+5k

455.

17x2+7x17x2+7x

456.

5c2+9c5c2+9c

457.

4q2+7q4q2+7q

458.

5p2+25p5p2+25p

459.

3r2+27r3r2+27r

460.

24q212q24q212q

461.

30u210u30u210u

462.

yz+4zyz+4z

463.

ab+8bab+8b

464.

60x6x360x6x3

465.

55y11y455y11y4

466.

48r412r348r412r3

467.

45c315c245c315c2

468.

4a34ab24a34ab2

469.

6c36cd26c36cd2

470.

30u3+80u230u3+80u2

471.

48x3+72x248x3+72x2

472.

120y6+48y4120y6+48y4

473.

144a6+90a3144a6+90a3

474.

4q2+24q+284q2+24q+28

475.

10y2+50y+4010y2+50y+40

476.

15z230z9015z230z90

477.

12u236u10812u236u108

478.

3a424a3+18a23a424a3+18a2

479.

5p420p315p25p420p315p2

480.

11x6+44x5121x411x6+44x5121x4

481.

8c5+40c456c38c5+40c456c3

482.

−3n24−3n24

483.

−7p84−7p84

484.

−15a240a−15a240a

485.

−18b266b−18b266b

486.

−10y3+60y2−10y3+60y2

487.

−8a3+32a2−8a3+32a2

488.

−4u5+56u3−4u5+56u3

489.

−9b5+63b3−9b5+63b3

Everyday Math

490.

Revenue A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of pp dollars each is given by the polynomial −5p2+150p.−5p2+150p. Factor the greatest common factor from this polynomial.

491.

Height of a baseball The height of a baseball hit with velocity 8080 feet/second at 44 feet above ground level is −16t2+80t+4,−16t2+80t+4, with t=t= the number of seconds since it was hit. Factor the greatest common factor from this polynomial.

Writing Exercises

492.

The greatest common factor of 3636 and 6060 is 12.12. Explain what this means.

493.

What is the GCF of y4y4, y5y5, and y10y10? Write a general rule that tells how to find the GCF of yaya, ybyb, and ycyc.

Self Check

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

.

Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

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