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Elementary Algebra 2e

7.5 General Strategy for Factoring Polynomials

Elementary Algebra 2e7.5 General Strategy for Factoring Polynomials
  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:
  • Recognize and use the appropriate method to factor a polynomial completely
Be Prepared 7.15

Before you get started, take this readiness quiz.

Factor y22y24y22y24.
If you missed this problem, review Example 7.23.

Be Prepared 7.16

Factor 3t2+17t+103t2+17t+10.
If you missed this problem, review Example 7.38.

Be Prepared 7.17

Factor 36p260p+2536p260p+25.
If you missed this problem, review Example 7.42.

Be Prepared 7.18

Factor 5x2805x280.
If you missed this problem, review Example 7.52.

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. Figure 7.4 outlines a strategy you should use when factoring polynomials.

This figure presents a general strategy for factoring polynomials. First, at the top, there is GCF, which is where factoring starts. Below this, there are three options, binomial, trinomial, and more than three terms. For binomial, there are the difference of two squares, the sum of squares, the sum of cubes, and the difference of cubes. For trinomials, there are two forms, x squared plus bx plus c and ax squared 2 plus b x plus c. There are also the sum and difference of two squares formulas as well as the “a c” method. Finally, for more than three terms, the method is grouping.
Figure 7.4

How To

Factor polynomials.

  1. Step 1. Is there a greatest common factor?
    • Factor it out.
  2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?
    • If it is a binomial:
      Is it a sum?
      • Of squares? Sums of squares do not factor.
      • Of cubes? Use the sum of cubes pattern.
      Is it a difference?
      • Of squares? Factor as the product of conjugates.
      • Of cubes? Use the difference of cubes pattern.
    • If it is a trinomial:
      Is it of the form x2+bx+cx2+bx+c? Undo FOIL.
      Is it of the form ax2+bx+cax2+bx+c?
      • If aa and cc are squares, check if it fits the trinomial square pattern.
      • Use the trial and error or “ac” method.
    • If it has more than three terms:
      Use the grouping method.
  3. Step 3. Check.
    • Is it factored completely?
    • Do the factors multiply back to the original polynomial?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example 7.59

Factor completely: 4x5+12x44x5+12x4.

Try It 7.117

Factor completely: 3a4+18a33a4+18a3.

Try It 7.118

Factor completely: 45b6+27b545b6+27b5.

Example 7.60

Factor completely: 12x211x+212x211x+2.

Try It 7.119

Factor completely: 10a217a+610a217a+6.

Try It 7.120

Factor completely: 8x218x+98x218x+9.

Example 7.61

Factor completely: g3+25gg3+25g.

Try It 7.121

Factor completely: x3+36xx3+36x.

Try It 7.122

Factor completely: 27y2+4827y2+48.

Example 7.62

Factor completely: 12y27512y275.

Try It 7.123

Factor completely: 16x336x16x336x.

Try It 7.124

Factor completely: 27y24827y248.

Example 7.63

Factor completely: 4a212ab+9b24a212ab+9b2.

Try It 7.125

Factor completely: 4x2+20xy+25y24x2+20xy+25y2.

Try It 7.126

Factor completely: 9m2+42mn+49n29m2+42mn+49n2.

Example 7.64

Factor completely: 6y218y606y218y60.

Try It 7.127

Factor completely: 8y2+16y248y2+16y24.

Try It 7.128

Factor completely: 5u215u2705u215u270.

Example 7.65

Factor completely: 24x3+8124x3+81.

Try It 7.129

Factor completely: 250m3+432250m3+432.

Try It 7.130

Factor completely: 81q3+19281q3+192.

Example 7.66

Factor completely: 2x4322x432.

Try It 7.131

Factor completely: 4a4644a464.

Try It 7.132

Factor completely: 7y477y47.

Example 7.67

Factor completely: 3x2+6bx3ax6ab3x2+6bx3ax6ab.

Try It 7.133

Factor completely: 6x212xc+6bx12bc6x212xc+6bx12bc.

Try It 7.134

Factor completely: 16x2+24xy4x6y16x2+24xy4x6y.

Example 7.68

Factor completely: 10x234x2410x234x24.

Try It 7.135

Factor completely: 4p216p+124p216p+12.

Try It 7.136

Factor completely: 6q29q66q29q6.

Section 7.5 Exercises

Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

279.

10x4+35x310x4+35x3

280.

18p6+24p318p6+24p3

281.

y2+10y39y2+10y39

282.

b217b+60b217b+60

283.

2n2+13n72n2+13n7

284.

8x29x38x29x3

285.

a5+9a3a5+9a3

286.

75m3+12m75m3+12m

287.

121r2s2121r2s2

288.

49b236a249b236a2

289.

8m2328m232

290.

36q210036q2100

291.

25w260w+3625w260w+36

292.

49b2112b+6449b2112b+64

293.

m2+14mn+49n2m2+14mn+49n2

294.

64x2+16xy+y264x2+16xy+y2

295.

7b2+7b427b2+7b42

296.

3n2+30n+723n2+30n+72

297.

3x3813x381

298.

5t3405t340

299.

k416k416

300.

m481m481

301.

15pq15p+12q1215pq15p+12q12

302.

12ab6a+10b512ab6a+10b5

303.

4x2+40x+844x2+40x+84

304.

5q215q905q215q90

305.

u5+u2u5+u2

306.

5n3+3205n3+320

307.

4c2+20cd+81d24c2+20cd+81d2

308.

25x2+35xy+49y225x2+35xy+49y2

309.

10m4625010m46250

310.

3v47683v4768

Everyday Math

311.

Watermelon drop A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.

  1. The binomial −16t2+80−16t2+80 gives the height of the watermelon tt seconds after it is dropped. Factor the greatest common factor from this binomial.
  2. If the watermelon is thrown down with initial velocity 8 feet per second, its height after tt seconds is given by the trinomial −16t28t+80−16t28t+80. Completely factor this trinomial.
312.

Pumpkin drop A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall.

  1. The binomial −16t2+128−16t2+128 gives the height of the pumpkin t seconds after it is dropped. Factor the greatest common factor from this binomial.
  2. If the pumpkin is thrown down with initial velocity 32 feet per second, its height after tt seconds is given by the trinomial −16t232t+128−16t232t+128. Completely factor this trinomial.

Writing Exercises

313.

The difference of squares y4625y4625 can be factored as (y225)(y2+25)(y225)(y2+25). But it is not completely factored. What more must be done to completely factor it?

314.

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

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 row states “recognize and use the appropriate method to factor a polynomial completely”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

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

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