Skip to Content
OpenStax Logo
Elementary Algebra

7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1

Elementary Algebra7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
  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 Solve 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 Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    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
    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 a preliminary strategy to factor polynomials completely
  • Factor trinomials of the form ax2+bx+cax2+bx+c with a GCF
  • Factor trinomials using trial and error
  • Factor trinomials using the ‘ac’ method
Be Prepared 7.3

Before you get started, take this readiness quiz.

  1. Find the GCF of 45p2and30p645p2and30p6.
    If you missed this problem, review Example 7.2.
  2. Multiply (3y+4)(2y+5)(3y+4)(2y+5).
    If you missed this problem, review Example 6.40.
  3. Combine like terms 12x2+3x+5x+912x2+3x+5x+9.
    If you missed this problem, review Example 1.24.

Recognize a Preliminary Strategy for Factoring

Let’s summarize where we are so far with factoring polynomials. In the first two sections of this chapter, we used three methods of factoring: factoring the GCF, factoring by grouping, and factoring a trinomial by “undoing” FOIL. More methods will follow as you continue in this chapter, as well as later in your studies of algebra.

How will you know when to use each factoring method? As you learn more methods of factoring, how will you know when to apply each method and not get them confused? It will help to organize the factoring methods into a strategy that can guide you to use the correct method.

As you start to factor a polynomial, always ask first, “Is there a greatest common factor?” If there is, factor it first.

The next thing to consider is the type of polynomial. How many terms does it have? Is it a binomial? A trinomial? Or does it have more than three terms?

If it is a trinomial where the leading coefficient is one, x2+bx+cx2+bx+c, use the “undo FOIL” method.

If it has more than three terms, try the grouping method. This is the only method to use for polynomials of more than three terms.

Some polynomials cannot be factored. They are called “prime.”

Below we summarize the methods we have so far. These are detailed in Choose a strategy to factor polynomials completely.

This figure lists strategies for factoring polynomials. At the top of the figure is G C F, where factoring always starts. From there, the figure has three branches. The first is binomial, the second is trinomial with the form x ^ 2 + b x +c, and the third is “more than three terms”, which is labeled with grouping.

How To

Choose a strategy to factor polynomials completely.

  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, right now we have no method to factor it.
    • If it is a trinomial of the form x2+bx+cx2+bx+c: Undo FOIL (x)(x)(x)(x)
    • If it has more than three terms: Use the grouping method.
  3. Step 3. Check by multiplying the factors.

Use the preliminary strategy to completely factor a polynomial. A polynomial is factored completely if, other than monomials, all of its factors are prime.

Example 7.29

Identify the best method to use to factor each polynomial.

  1. 6y2726y272
  2. r210r24r210r24
  3. p2+5p+pq+5qp2+5p+pq+5q
Try It 7.57

Identify the best method to use to factor each polynomial:

  1. 4y2+324y2+32
  2. y2+10y+21y2+10y+21
  3. yz+2y+3z+6yz+2y+3z+6
Try It 7.58

Identify the best method to use to factor each polynomial:

  1. ab+a+4b+4ab+a+4b+4
  2. 3k2+153k2+15
  3. p2+9p+8p2+9p+8

Factor Trinomials of the form ax2 + bx + c with a GCF

Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form ax2+bx+cax2+bx+c.

Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods in the last section. Let’s do a few examples to see how this works.

Watch out for the signs in the next two examples.

Example 7.30

Factor completely: 2n28n422n28n42.

Try It 7.59

Factor completely: 4m24m84m24m8.

Try It 7.60

Factor completely: 5k215k505k215k50.

Example 7.31

Factor completely: 4y236y+564y236y+56.

Try It 7.61

Factor completely: 3r29r+63r29r+6.

Try It 7.62

Factor completely: 2t210t+122t210t+12.

In the next example the GCF will include a variable.

Example 7.32

Factor completely: 4u3+16u220u4u3+16u220u.

Try It 7.63

Factor completely: 5x3+15x220x5x3+15x220x.

Try It 7.64

Factor completely: 6y3+18y260y6y3+18y260y.

Factor Trinomials using Trial and Error

What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.

Let’s factor the trinomial 3x2+5x+23x2+5x+2.

From our earlier work we expect this will factor into two binomials.

3x2+5x+2()()3x2+5x+2()()

We know the first terms of the binomial factors will multiply to give us 3x23x2. The only factors of 3x23x2 are 1x,3x1x,3x. We can place them in the binomials.

This figure has the polynomial 3 x^ 2 +5 x +2. Underneath there are two terms, 1 x, and 3 x. Below these are the two factors x and (3 x) being shown multiplied.

Check. Does 1x·3x=3x21x·3x=3x2?

We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1 and 2. But we now have two cases to consider as it will make a difference if we write 1, 2, or 2, 1.

This figure demonstrates the possible factors of the polynomial 3x^2 +5x +2. The polynomial is written twice. Underneath both, there are the terms 1x, 3x under the 3x^2. Also, there are the factors 1,2 under the 2 term. At the bottom of the figure there are two possible factorizations of the polynomial. The first is (x + 1)(3x + 2) and the next is (x + 2)(3x + 1).

Which factors are correct? To decide that, we multiply the inner and outer terms.

This figure demonstrates the possible factors of the polynomial 3 x^ 2 + 5 x +2. The polynomial is written twice. Underneath both, there are the terms 1 x, 3 x under the 3 x ^ 2. Also, there are the factors 1, 2 under the 2 term. At the bottom of the figure there are two possible factorizations of the polynomial. The first is (x + 1)(3 x + 2). Underneath this factorization are the products 3 x from multiplying the middle terms 1 and 3 x. Also there is the product of 2 x from multiplying the outer terms x and 2. These products of 3 x and 2 x add to 5 x. Underneath the second factorization are the products 6 x from multiplying the middle terms 2 and 3 x. Also there is the product of 1 x from multiplying the outer terms x and 1. These two products of 6 x and 1 x add to 7 x.

Since the middle term of the trinomial is 5x, the factors in the first case will work. Let’s FOIL to check.

(x+1)(3x+2)3x2+2x+3x+23x2+5x+2(x+1)(3x+2)3x2+2x+3x+23x2+5x+2

Our result of the factoring is:

3x2+5x+2(x+1)(3x+2)3x2+5x+2(x+1)(3x+2)

Example 7.33

How to Factor Trinomials of the Form ax2+bx+cax2+bx+c Using Trial and Error

Factor completely: 3y2+22y+73y2+22y+7.

Try It 7.65

Factor completely: 2a2+5a+32a2+5a+3.

Try It 7.66

Factor completely: 4b2+5b+14b2+5b+1.

How To

Factor trinomials of the form ax2+bx+cax2+bx+c using trial and error.

  1. Step 1. Write the trinomial in descending order of degrees.
  2. Step 2. Find all the factor pairs of the first term.
  3. Step 3. Find all the factor pairs of the third term.
  4. Step 4. Test all the possible combinations of the factors until the correct product is found.
  5. Step 5. Check by multiplying.

When the middle term is negative and the last term is positive, the signs in the binomials must both be negative.

Example 7.34

Factor completely: 6b213b+56b213b+5.

Try It 7.67

Factor completely: 8x213x+38x213x+3.

Try It 7.68

Factor completely: 10y237y+710y237y+7.

When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.

Example 7.35

Factor completely: 14x247x714x247x7.

Try It 7.69

Factor completely: 8a23a58a23a5.

Try It 7.70

Factor completely: 6b2b156b2b15.

Example 7.36

Factor completely: 18n237n+1518n237n+15.

Try It 7.71

Factor completely: 18x23x1018x23x10.

Try It 7.72

Factor completely: 30y253y2130y253y21.

Don’t forget to look for a GCF first.

Example 7.37

Factor completely: 10y4+55y3+60y210y4+55y3+60y2.

Try It 7.73

Factor completely: 15n385n2+100n15n385n2+100n.

Try It 7.74

Factor completely: 56q3+320q296q56q3+320q296q.

Factor Trinomials using the “ac” Method

Another way to factor trinomials of the form ax2+bx+cax2+bx+c is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!

Example 7.38

How to Factor Trinomials Using the “ac” Method

Factor: 6x2+7x+26x2+7x+2.

Try It 7.75

Factor: 6x2+13x+26x2+13x+2.

Try It 7.76

Factor: 4y2+8y+34y2+8y+3.

How To

Factor trinomials of the form using the “ac” method.

  1. Step 1. Factor any GCF.
  2. Step 2. Find the product ac.
  3. Step 3. Find two numbers m and n that:
    Multiply toacm·n=a·cAdd tobm+n=bMultiply toacm·n=a·cAdd tobm+n=b
  4. Step 4. Split the middle term using m and n: This figure shows two equations. The top equation reads a times x squared plus b times x plus c. Under this, is the equation a times x squared plus m times x plus n times x plus c. Above the m times x plus n times x is a bracket with b times x above it.
  5. Step 5. Factor by grouping.
  6. Step 6. Check by multiplying the factors.

When the third term of the trinomial is negative, the factors of the third term will have opposite signs.

Example 7.39

Factor: 8u217u218u217u21.

Try It 7.77

Factor: 20h2+13h1520h2+13h15.

Try It 7.78

Factor: 6g2+19g206g2+19g20.

Example 7.40

Factor: 2x2+6x+52x2+6x+5.

Try It 7.79

Factor: 10t2+19t1510t2+19t15.

Try It 7.80

Factor: 3u2+8u+53u2+8u+5.

Don’t forget to look for a common factor!

Example 7.41

Factor: 10y255y+7010y255y+70.

Try It 7.81

Factor: 16x232x+1216x232x+12.

Try It 7.82

Factor: 18w239w+1818w239w+18.

We can now update the Preliminary Factoring Strategy, as shown in Figure 7.2 and detailed in Choose a strategy to factor polynomials completely (updated), to include trinomials of the form ax2+bx+cax2+bx+c. Remember, some polynomials are prime and so they cannot be factored.

This figure has the strategy for factoring polynomials. At the top of the figure is GCF. Below this, there are three options. The first is binomial. The second is trinomial. Under trinomial there are x squared + b x + c and a x squared + b x +c. The two methods here are trial and error and the “a c” method. The third option is for more than three terms. It is grouping.
Figure 7.2

How To

Choose a strategy to factor polynomials completely (updated).

  1. Step 1. Is there a greatest common factor?
    • Factor it.
  2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?
    • If it is a binomial, right now we have no method to factor it.
    • If it is a trinomial of the form x2+bx+cx2+bx+c
      Undo FOIL (x)(x)(x)(x).
    • If it is a trinomial of the form ax2+bx+cax2+bx+c
      Use Trial and Error or the “ac” method.
    • If it has more than three terms
      Use the grouping method.
  3. Step 3. Check by multiplying the factors.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with factoring trinomials of the form ax2+bx+cax2+bx+c.

Section 7.3 Exercises

Practice Makes Perfect

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

135.
  1. 10q2+5010q2+50
  2. a25a14a25a14
  3. uv+2u+3v+6uv+2u+3v+6
136.
  1. n2+10n+24n2+10n+24
  2. 8u2+168u2+16
  3. pq+5p+2q+10pq+5p+2q+10
137.
  1. x2+4x21x2+4x21
  2. ab+10b+4a+40ab+10b+4a+40
  3. 6c2+246c2+24
138.
  1. 20x2+10020x2+100
  2. uv+6u+4v+24uv+6u+4v+24
  3. y28y+15y28y+15

Factor Trinomials of the form ax2+bx+cax2+bx+c with a GCF

In the following exercises, factor completely.

139.

5x2+35x+305x2+35x+30

140.

12s2+24s+1212s2+24s+12

141.

2z22z242z22z24

142.

3u212u363u212u36

143.

7v263v+567v263v+56

144.

5w230w+455w230w+45

145.

p38p220pp38p220p

146.

q35q224qq35q224q

147.

3m321m2+30m3m321m2+30m

148.

11n355n2+44n11n355n2+44n

149.

5x4+10x375x25x4+10x375x2

150.

6y4+12y348y26y4+12y348y2

Factor Trinomials Using Trial and Error

In the following exercises, factor.

151.

2t2+7t+52t2+7t+5

152.

5y2+16y+115y2+16y+11

153.

11x2+34x+311x2+34x+3

154.

7b2+50b+77b2+50b+7

155.

4w25w+14w25w+1

156.

5x217x+65x217x+6

157.

6p219p+106p219p+10

158.

21m229m+1021m229m+10

159.

4q27q24q27q2

160.

10y253y1110y253y11

161.

4p2+17p154p2+17p15

162.

6u2+5u146u2+5u14

163.

16x232x+1616x232x+16

164.

81a2+153a1881a2+153a18

165.

30q3+140q2+80q30q3+140q2+80q

166.

5y3+30y235y5y3+30y235y

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

167.

5n2+21n+45n2+21n+4

168.

8w2+25w+38w2+25w+3

169.

9z2+15z+49z2+15z+4

170.

3m2+26m+483m2+26m+48

171.

4k216k+154k216k+15

172.

4q29q+54q29q+5

173.

5s29s+45s29s+4

174.

4r220r+254r220r+25

175.

6y2+y156y2+y15

176.

6p2+p226p2+p22

177.

2n227n452n227n45

178.

12z241z1112z241z11

179.

3x2+5x+43x2+5x+4

180.

4y2+15y+64y2+15y+6

181.

60y2+290y5060y2+290y50

182.

6u246u166u246u16

183.

48z3102z245z48z3102z245z

184.

90n3+42n2216n90n3+42n2216n

185.

16s2+40s+2416s2+40s+24

186.

24p2+160p+9624p2+160p+96

187.

48y2+12y3648y2+12y36

188.

30x2+105x6030x2+105x60

Mixed Practice

In the following exercises, factor.

189.

12y229y+1412y229y+14

190.

12x2+36y24z12x2+36y24z

191.

a2a20a2a20

192.

m2m12m2m12

193.

6n2+5n46n2+5n4

194.

12y237y+2112y237y+21

195.

2p2+4p+32p2+4p+3

196.

3q2+6q+23q2+6q+2

197.

13z2+39z2613z2+39z26

198.

5r2+25r+305r2+25r+30

199.

x2+3x28x2+3x28

200.

6u2+7u56u2+7u5

201.

3p2+21p3p2+21p

202.

7x221x7x221x

203.

6r2+30r+366r2+30r+36

204.

18m2+15m+318m2+15m+3

205.

24n2+20n+424n2+20n+4

206.

4a2+5a+24a2+5a+2

207.

x2+2x24x2+2x24

208.

2b27b+42b27b+4

Everyday Math

209.

Height of a toy rocket The height of a toy rocket launched with an initial speed of 80 feet per second from the balcony of an apartment building is related to the number of seconds, t, since it is launched by the trinomial −16t2+80t+96−16t2+80t+96. Completely factor the trinomial.

210.

Height of a beach ball The height of a beach ball tossed up with an initial speed of 12 feet per second from a height of 4 feet is related to the number of seconds, t, since it is tossed by the trinomial −16t2+12t+4−16t2+12t+4. Completely factor the trinomial.

Writing Exercises

211.

List, in order, all the steps you take when using the “ac” method to factor a trinomial of the form ax2+bx+c.ax2+bx+c.

212.

How is the “ac” method similar to the “undo FOIL” method? How is it different?

213.

What are the questions, in order, that you ask yourself as you start to factor a polynomial? What do you need to do as a result of the answer to each question?

214.

On your paper draw the chart that summarizes the factoring strategy. Try to do it without looking at the book. When you are done, look back at the book to finish it or verify it.

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 first row is “recognize a preliminary strategy to factor polynomials completely”. The second row is “factor trinomials of the form a x ^ 2 + b x + c with a GCF”. The third row is “factor trinomials using trial and error”. And the fourth row is “factor trinomials using the “ac” method”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 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/elementary-algebra/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/elementary-algebra/pages/1-introduction
Citation information

© Oct 23, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 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.