Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Intermediate Algebra 2e

6.4 General Strategy for Factoring Polynomials

Intermediate Algebra 2e6.4 General Strategy for Factoring Polynomials

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.

2 n 2 + 13 n 7 2 n 2 + 13 n 7

234.

8 x 2 9 x 3 8 x 2 9 x 3

235.

a 5 + 9 a 3 a 5 + 9 a 3

236.

75 m 3 + 12 m 75 m 3 + 12 m

237.

121 r 2 s 2 121 r 2 s 2

238.

49 b 2 36 a 2 49 b 2 36 a 2

239.

8 m 2 32 8 m 2 32

240.

36 q 2 100 36 q 2 100

241.

25 w 2 60 w + 36 25 w 2 60 w + 36

242.

49 b 2 112 b + 64 49 b 2 112 b + 64

243.

m 2 + 14 m n + 49 n 2 m 2 + 14 m n + 49 n 2

244.

64 x 2 + 16 x y + y 2 64 x 2 + 16 x y + y 2

245.

7 b 2 + 7 b 42 7 b 2 + 7 b 42

246.

30 n 2 + 30 n + 72 30 n 2 + 30 n + 72

247.

3 x 4 y 81 x y 3 x 4 y 81 x y

248.

4 x 5 y 32 x 2 y 4 x 5 y 32 x 2 y

249.

k 4 16 k 4 16

250.

m 4 81 m 4 81

251.

5 x 5 y 2 80 x y 2 5 x 5 y 2 80 x y 2

252.

48 x 5 y 2 243 x y 2 48 x 5 y 2 243 x y 2

253.

15 p q 15 p + 12 q 12 15 p q 15 p + 12 q 12

254.

12 a b 6 a + 10 b 5 12 a b 6 a + 10 b 5

255.

4 x 2 + 40 x + 84 4 x 2 + 40 x + 84

256.

5 q 2 15 q 90 5 q 2 15 q 90

257.

4 u 5 + 4 u 2 v 3 4 u 5 + 4 u 2 v 3

258.

5 m 4 n + 320 m n 4 5 m 4 n + 320 m n 4

259.

4 c 2 + 20 c d + 81 d 2 4 c 2 + 20 c d + 81 d 2

260.

25 x 2 + 35 x y + 49 y 2 25 x 2 + 35 x y + 49 y 2

261.

10 m 4 6250 10 m 4 6250

262.

3 v 4 768 3 v 4 768

263.

36 x 2 y + 15 x y 6 y 36 x 2 y + 15 x y 6 y

264.

60 x 2 y 75 x y + 30 y 60 x 2 y 75 x y + 30 y

265.

8 x 3 27 y 3 8 x 3 27 y 3

266.

64 x 3 + 125 y 3 64 x 3 + 125 y 3

267.

y 6 1 y 6 1

268.

y 6 + 1 y 6 + 1

269.

9 x 2 6 x y + y 2 49 9 x 2 6 x y + y 2 49

270.

16 x 2 24 x y + 9 y 2 64 16 x 2 24 x y + 9 y 2 64

271.

( 3 x + 1 ) 2 6 ( 3 x + 1 ) + 9 ( 3 x + 1 ) 2 6 ( 3 x + 1 ) + 9

272.

( 4 x 5 ) 2 7 ( 4 x 5 ) + 12 ( 4 x 5 ) 2 7 ( 4 x 5 ) + 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?

Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
Citation information

© Jan 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.