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Intermediate Algebra

6.4 General Strategy for Factoring Polynomials

Intermediate Algebra6.4 General Strategy for Factoring Polynomials
  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:
  • Recognize and use the appropriate method to factor a polynomial completely

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. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.

General Strategy for Factoring Polynomials

This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.

How To

Use a general strategy for factoring 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+c?x2+bx+c? Undo FOIL.
    • Is it of the form ax2+bx+c?ax2+bx+c?
      If a and c 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 6.35

Factor completely: 7x321x270x.7x321x270x.

Try It 6.69

Factor completely: 8y3+16y224y.8y3+16y224y.

Try It 6.70

Factor completely: 5y315y2270y.5y315y2270y.

Be careful when you are asked to factor a binomial as there are several options!

Example 6.36

Factor completely: 24y2150.24y2150.

Try It 6.71

Factor completely: 16x336x.16x336x.

Try It 6.72

Factor completely: 27y248.27y248.

The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.

Example 6.37

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

Try It 6.73

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

Try It 6.74

Factor completely: 9x224xy+16y2.9x224xy+16y2.

Remember, sums of squares do not factor, but sums of cubes do!

Example 6.38

Factor completely 12x3y2+75xy2.12x3y2+75xy2.

Try It 6.75

Factor completely: 50x3y+72xy.50x3y+72xy.

Try It 6.76

Factor completely: 27xy3+48xy.27xy3+48xy.

When using the sum or difference of cubes pattern, being careful with the signs.

Example 6.39

Factor completely: 24x3+81y3.24x3+81y3.

Try It 6.77

Factor completely: 250m3+432n3.250m3+432n3.

Try It 6.78

Factor completely: 2p3+54q3.2p3+54q3.

Example 6.40

Factor completely: 3x5y48xy.3x5y48xy.

Try It 6.79

Factor completely: 4a5b64ab.4a5b64ab.

Try It 6.80

Factor completely: 7xy57xy.7xy57xy.

Example 6.41

Factor completely: 4x2+8bx4ax8ab.4x2+8bx4ax8ab.

Try It 6.81

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

Try It 6.82

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

Taking out the complete GCF in the first step will always make your work easier.

Example 6.42

Factor completely: 40x2y+44xy24y.40x2y+44xy24y.

Try It 6.83

Factor completely: 4p2q16pq+12q.4p2q16pq+12q.

Try It 6.84

Factor completely: 6pq29pq6p.6pq29pq6p.

When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.

Example 6.43

Factor completely: 9x212xy+4y249.9x212xy+4y249.

Try It 6.85

Factor completely: 4x212xy+9y225.4x212xy+9y225.

Try It 6.86

Factor completely: 16x224xy+9y264.16x224xy+9y264.

Section 6.4 Exercises

Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

233.

2n2+13n72n2+13n7

234.

8x29x38x29x3

235.

a5+9a3a5+9a3

236.

75m3+12m75m3+12m

237.

121r2s2121r2s2

238.

49b236a249b236a2

239.

8m2328m232

240.

36q210036q2100

241.

25w260w+3625w260w+36

242.

49b2112b+6449b2112b+64

243.

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

244.

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

245.

7b2+7b427b2+7b42

246.

30n2+30n+7230n2+30n+72

247.

3x4y81xy3x4y81xy

248.

4x5y32x2y4x5y32x2y

249.

k416k416

250.

m481m481

251.

5x5y280xy25x5y280xy2

252.

48x5y2243xy248x5y2243xy2

253.

15pq15p+12q1215pq15p+12q12

254.

12ab6a+10b512ab6a+10b5

255.

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

256.

5q215q905q215q90

257.

4u5+4u2v34u5+4u2v3

258.

5m4n+320mn45m4n+320mn4

259.

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

260.

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

261.

10m4625010m46250

262.

3v47683v4768

263.

36x2y+15xy6y36x2y+15xy6y

264.

60x2y75xy+30y60x2y75xy+30y

265.

8x327y38x327y3

266.

64x3+125y364x3+125y3

267.

y61y61

268.

y6+1y6+1

269.

9x26xy+y2499x26xy+y249

270.

16x224xy+9y26416x224xy+9y264

271.

(3x+1)26(3x+1)+9(3x+1)26(3x+1)+9

272.

(4x5)27(4x5)+12(4x5)27(4x5)+12

Writing Exercises

273.

Explain what it mean to factor a polynomial completely.

274.

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.

275.

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.

276.

Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.

Self Check

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

This table has 4 columns, 1 row 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 statement: recognize and use the appropriate method to factor a polynomial completely. The remaining columns are blank.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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